Are Newton's "laws" of motion laws or definitions of force and mass? If you consider them as laws, then there must be independent definitions of force and mass but I don't think there's such definitions.
If you consider them as definitions, then why are they still called laws?
 A: I think the answer by Joshphysics is very good. In particular the statement that asserting existence is a key element.
The idea is to restate the laws of motion in such a way that the question law versus definition issue becomes clearer. 
In analogy with thermodynamics I will state a 'law zero'; a law that comes before the historical 'First law'. 
As with Joshphysics's answer the following treatment is for the Newtonian domain.
Law zero: 
(Assertion of existence) 
There exists opposition to change of an object's velocity. This opposition to change of velocity is called 'inertia'. 
First law: 
(The uniformity law) 
The opposition to change of velocity is uniform in all positions in space and in all spatial directions. 
Second law: 
(The acceleration law) 
The change of velocity is proportional to the exerted force, and inversely proportional to the mass.
The above statements are not definitions. 
For comparison, the zero point of the Celcius scale is a definition; it is interexchangeable with another definition of zero point of temperature scale. The laws of motion are not exchangeable for other statements.
The concept of force is also applicable in statics, hence Force can also be defined in the context of a static case (compression), and then we check for consistency with Force defined in terms of dynamics. As we know: we find consistency.
For mass things are more interesting. Mass is in fact defined by the laws of motion. Trivial example: if you would use the volume of an object as a measure of its mass the second law would not apply universally. It's the law of motion that singles out what an object's mass is: precisely that property for which the second law holds good.
The lesson is that if you would insist that any statement is either a physics law, or a definition, you would totally bog yourself down.
Our physics laws are both: they are statements about inherent properties of Nature, and they define the concepts that the laws are valid for.



Additional remarks:
The first and second law together are sufficient to imply the historical third law. This can be recognized in the following way:
Let object A and object B both be floating in space, not attached to any larger mass. 
From an abstract point of view it might be argued: there is a difference between:
Case 1: object A exerting a force upon object B, but B not on A 
Case 2: object A and object B exerting a force upon each other.
According to the laws of motion the above distinction is moot. Observationally the two cases are identical, making it meaningless to distinguish between them on an abstract level.
Assume for argument sake that object A exerts a attracting force upon object B, but B not upon A. Both A and B are floating in space. The leverage that object A has to pull object B towards itself is A's own inertia. A has no other leverage, A is not attached to any larger mass. A can pull B closer to itself if and only if A is itself in acceleration towards B. There is no scenario, no observation, where Case 1 and Case 2 are distinguishable, hence Case 1 and Case 2 must be regarded as one and the same case.


The first law and second law together are sufficient to imply the superposition of forces.
A: joshphysics's answer is excellent, and a perfectly good logical ordering of concepts, in which force is defined in terms of mass.  I personally prefer a slightly different logical ordering (which of course ends up being equivalent), in which mass is defined in terms of force:
First law: Local inertial reference frames exist.
I can't improve on joshphysics's excellent explanation here.
Second law: Every object's mass exists, and is independent of the force applied to it.
We define a "force" $F_i$ to be a physical influence arising from a repeatable experimental setup.  ($i$ is just a label, not a vector component.)  For example, we could consider a single rubber band, stretched by a fixed amount, to which we connect a series of different "test objects."  This defines a force $F_1$ which is not a vector quantity (hence the lack of bold script), but instead a label for a particular experimental setup.  Or we could consider the gravitational pull $F_2$ from Jupiter on various "test objects" when it is at a particular location and distance relative to the test object. A given force $F_i$ acting on a given test object $o_j$ will impart on it a measurable acceleration vector ${\bf a}(F_i, o_j)$.
Now we find three nontrivial empirical results:
(i) If forces $F_1$ and $F_2$ induce accelerations ${\bf a}_1$ and ${\bf a}_2$ in an object when applied individually, then they induce acceleration ${\bf a}_1 + {\bf a}_2$ in the object when applied simultaneously.
(ii) A given force $F_i$ accelerates all test objects in the same direction (although with different magnitudes).  In other words,
$${\bf a}(F_i, o_j) \parallel {\bf a}(F_i, o_{j'})$$
for all $i$, $j$, and $j'$.
(iii) Suppose we have two different forces $F_1$ and $F_2$ (e.g. two rubber bands of different stiffness) and two different test objects $o_A$ and $o_B$.  The following equality always holds:
$$\frac{|{\bf a}(F_1, o_A)|}{|{\bf a}(F_1, o_B)|} = \frac{|{\bf a}(F_2, o_A)|}{|{\bf a}(F_2, o_B)|}.$$
This suggests a natural way to systematically quantify the effects of the various forces.  First take a particular test object $O$ and assign to it an arbitrary scalar quantity $m_O$ called its "mass."  Don't worry about the physical significance of this quantity yet.  Note that only this one particular object has a well-defined "mass" at this stage.  Now apply all of your different forces to the object $O$.  Each force $F_i$ will induce some acceleration ${\bf a}(F_i, O)$ on $O$.  Now assign to each force $F_i$ a vector quantity
$${\bf F}_i := m_O\, {\bf a}(F_i, O)$$
which "records" its action on the test object $O$.  Note that Newton's second law is trivially true only for the particular test object $O$.  Also note that changing the value of $m_O$ simply dilates all the force vectors by the same amount, so you might as well just choose mass units in which it has the numerical value of $1$.  The empirical observation (ii) above can now be rephrased as
(ii') For all forces $F_i$ and test objects $o_j$,
$${\bf F}_i \parallel {\bf a}(F_i, o_j).$$
We can therefore define a scalar quantity $m_{(i,j)}$, which depends both on the applied force and on the test object, such that
$${\bf F}_i = m_{(i,j)} {\bf a}(F_i, o_j).$$
This justifies the first claim of the Second Law, that every object's mass exists.  Recall from the definition of the force vector that
$$m_O {\bf a}(F_i, O) = m_{(i,j)} {\bf a}(F_i, o_j),$$ so only the ratio $m_{(i,j)} / m_O$ is physically measurable, as mentioned above.
If we let $o_B$ be the test object $O$, then empirical observation (iii) above can be rearranged to $m_{(1,A)} = m_{(2,A)}$ for all test objects $o_A$, justifying the second claim of the Second Law that an object's mass does not depend on the external force applied to it.
Finally, the facts that (a) induced accelerations add as vectors and (b) an object's mass does not depend on the applied force, together imply that applied forces add as vectors as well.
Third law: When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object.
We already defined the force vector ${\bf F}$ above, so this is clearly a nontrivial empirical observation rather than a definition.
A: In my view, standard statements of Newton's laws are usually overly concise, and this lack of detail causes confusion about what is a definition, and what is an empirical fact.  To avoid this confusion, let's proceed in a systematic way that makes the distinctions between these definitions and empirical statements clear.
What follows certainly is not the original statement of the laws made by Newton himself; it is a modern interpretation meant to clarify the foundations of Newtonian mechanics.  As a result, the laws will be presented out of order in the interest of logical clarity.
To start off, we note that the definitions of mass and force given below will require the concept of a local inertial frame.  These are frames of reference in which when an object is isolated from all other matter, it's local acceleration is zero.  It is an empirical fact that such frames exist, and we'll take this as the first law:
First Law. Local inertial reference frames exist.
How is this in any way related to the first law we know and love?  Well, the way it is often stated, it basically says "if an object isn't interacting with anything, then it won't accelerate."  Of course, this is not entirely correct since there are reference frames (non-inertial ones) in which this statement breaks down.  You could then say, all right, all we need to do then is to qualify this statement of the first law by saying "provided we are making observations in an inertial frame, an object that doesn't interact with anything won't accelerate," but one could then object that this merely follows from the definition of inertial frames, so it has no physical content.  However, going one step further, we see that it's not at all clear a priori that inertial frames even exist, so the assertion that they do exist does have (deep) physical content.  In fact, it seems to me that this existence statement is kind of the essence of how the first law should be thought because it basically is saying that there are these special frames in the real world, and if your are observing an isolated object in one of these frames, then it won't accelerate just as Newton says.  This version of the first law also avoids the usual criticism that the first law trivially follows from the second law.
Equipped with the first law as stated above, we can now define mass.  In doing so, we'll find it useful to have another physical fact.
Third Law.  If two objects, sufficiently isolated from interactions with other objects, are observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant.
How is this related to the usual statement of the third law?  Well, thinking a bit "meta" here to use terms that we haven't defined yet, note that the way the third law is usually stated is "when objects interact in an inertial frame, they exert forces on each other that are equal in magnitude, but opposite in direction."  If you couple this with the second law, then you obtain that the product of their respective masses and accelerations are equal up to sign; $m_1\mathbf a_1 = -m_2\mathbf a_2$.  The statement of the third law given in this treatment is equivalent to this, but it's just a way of saying it that avoids referring to the concepts of force and mass which we have not yet defined.
Now, we use the third law to define mass.  Let two objects $O_0$ and $O_1$ be given, and suppose that they are being observed from a local inertial frame.  By the third law above, the ratio of their accelerations is some constant $c_{01}$;
\begin{align}
  \frac{a_0}{a_1} = c_{01}
\end{align}
We define object $O_0$ to have mass $m_0$ (whatever value we wish, like 1 for example if we want the reference object to be our unit mass), and we define the mass of $O_1$ to be
\begin{align}
  m_1=-c_{01}m_0
\end{align}
In this way, every object's mass is defined in terms of the reference mass.
We are now ready to define force.  Suppose that we observe an object $O$ of mass $m$ from a local inertial frame, and suppose that it is not isolated; it is exposed to some interaction $I$ to which we would like to associate a "force."  We observe that in the presence of only this interaction, the mass $m$ accelerates, and we define the force $\mathbf F_{I}$ exerted by $I$ on $O$ to be the product of the object's mass and its observed acceleration $\mathbf a$;
\begin{align}
  \mathbf F_{I} \equiv m\mathbf a
\end{align}
In other words, we are defining the force exerted by a single interaction $I$ on some object of mass $m$ as the mass times acceleration that a given object would have if it were exposed only to that interaction in a local inertial frame.
Second Law. If an object $O$ of mass $m$ in a local inertial frame simultaneously experiences interactions $I_1, \dots, I_N$, and if $\mathbf F_{I_i}$ is the force that would be exerted on $O$ by $I_i$ if it were the only interaction, then the acceleration $\mathbf a$ of $O$ will satisfy the following equation:
\begin{align}
  \mathbf F_{I_1} + \cdots +\mathbf F_{I_N} = m \mathbf a
\end{align}
A: First, I want to say I find your question excellent! It is very important, for anyone who wants to call himself a physicist, to know the answer on your question. 
EVERY PHYSICAL QUANTITY must be defined through operation of measurement OR through mathematical relations to other physical quantities which are already defined through operations of measurements. Meaning, we must know how to measure a physical quantity (directly or indirectly).  
For example, we define velocity as time derivative of position vector, and this makes sense only if we know how to measure time and length. 
Time is "defined" as measurement of specific clock (which has some specific properties in every way independent of time - we cannot say our specific clock, which we want to use as instrument for measuring time, must have properties of ticking after same TIME interval). We call one tick of our specific clock one second. Then, duration of some process we are observing is measured by counting ticking of our clock. N ticking means process lasted N seconds. Of course, if that process did not occur at the same place, we must use more than one same (i.e. having same properties) specific clock. We must use two clocks, but then clocks must be synchronized (by some defined procedure e.g. using light signals). I just want to add that what I said does not mean that every laboratory should have same specific clocks. We just defined time that way. Once we have done it, we then use some other clock and compare it with our specific clock. If their ticking match we can also use other clock for measuring time and so on. 
Length is defined similarly. We take some stick which we call one meter. That stick cannot have properties of being constant length (i.e. rigid) because we want to define length using that stick (we do not want circular definitions), so we want that our stick have some specific properties independent of length (we want it to be at the same pressure, temperature etc.). Then length of some object is how much our specific sticks we have between ending points of that object (we must know how we attach our sticks to each other i.e. what is straight line and we also must know simultaneously where ending points are, but I do not want to talk further about spacetime). Suppose we have N sticks we say length is N meter long. Once we defined procedure we can use some other sticks or methods for measuring length as long as they give same results as our specific stick (which we can check by comparison). 
LAWS OF PHYSICS are mathematical relations between physical quantities and we discover them by method of observations (empirically). Law is correct if our experiment says so. If I cannot experimentally (I neglect here technology problems) check some mathematical statement, then that statement is nothing more than mathematical expression, it is not a physical law.
So, mass, as physical quantity, is defined through measurement. We have some specific weighing scale and some specific object which we call one kilogram. We put other object we want to measure on the one plate of scale and counting how much our specific objects we must put on the other plate so out weighing scale is balanced. We counted N, so our object has mass of N kilograms. We can check that mass is additive quantity i.e. if we put two same objects we see that mass is 2N kilograms etc. We can measure mass by using different apparatus as long as they give same result as our first device (which we used for definition of mass). 
Same story is applied when we want to measure force. We define one Newton, procedure of measuring etc. We check that force is vector, find some other ways to measure force (they only need to match our first way).  
Momentum is defined as product of mass and velocity and measured indirectly. 
Now we know how mass and force are measured we can further explore properties of them i.e. we can now look for some law (mathematical relations) connecting quantities of mass and force. And we found through observations that F=ma and now we can interpret mass as measure of inertia of body and force as how much we would push or pull some body, but that is not definition of mass and force. If we defined force as F=ma, then this relation is not a physical law and we do not know nothing yet about force expect that it is calculated as product of mass and acceleration. 
Of course, we defined mass and force so they would be related somehow because we experience this Newton law on daily basis and we have already knew some properties which we want force and mass to have. 
"The development of physics is progressive, and as the theories of external world become crystallised, we often tend to replace the elementary physical quantities defined through operations of measurement by theoretical quantities believed to have a more fundamental şignificance in the external world. Thus the vis viva mvv, which is immediately determinable by experiment, becomes replaced by a generalised energy, virtually defined by having a property of conservation; and our problem becomes inverted - we have not to discover the properties of thing which we have recognized in nature, but to discover how to recognize in nature a thing whose properties we have assigned." - Arthur Stanley Eddington - Mathematical theory of relativity
Conservation of momentum then becomes experimentally provable. If we defined mass through conservation of momentum (by measuring ratio of accelerations of two isolated body and calling one body 1kg), then we cannot to check if conservation of momentum is true, bacause it would not be a law, but a definition of mass.
NEWTON LAWS ARE LAWS!
The first Newton law is most complicated, because it is hard to know if our system really is inertial or not (general theory of relativity beautifully explains this problem). But we can, as Newton originally did, says that distant stars are inertial system and every system in uniform motion relative to them is also inertial and second and third laws are correct in them.
Answer of "joshphysics" is logically precise, but physically wrong. 
A: To understand what Newton's three Laws really are, one needs to consider the notion of momentum. Momentum $\vec{p}$ of a point particle is the product of its mass $m$ (which will be defined implicitly later) and its instantaneous velocity $\vec{V}$, so $\vec{p}:=m\vec{V}$. Also, $m \in \mathbb{R}_+$ mass units and $ m:=const $ (reasons are so that $ m$ characterises a particle and does not make vectors $\vec{V}$ and $\vec{p}$ point in a different directions). One also needs to consider the Law of Conservation of a Linear Momentum, which is the consequence of space translation symmetry (contrary to a pupular belief that it is the consequence of Newton's Laws). 
Now, let's talk about the Newton's Laws:
Newton's first and third laws: consequence of the Law of Conservation of a Linear Momentum, nothing more. 
Newton's second law: a definition of a force, $\sum \vec{F}:=\dot{\vec{p}}$ (which also yields the familiar $\sum \vec{F}=m\vec{a}$) 
Remark: a question about measuring masses of point particles may arise, so here is the answer. Consider a system of two point particles moving along the $ x $-axis towards each other. Law of Conservation of Linear Momentum states:
\begin{align}m_1 \left |\vec{V}_{11}  \right | - m_2 \left |\vec{V}_{21}  \right | = m_2 \left |\vec{V}_{22}  \right |-m_1 \left |\vec{V}_{12}  \right |\end{align}
Defining $ m_1 $, for example, to be equal to one unit of mass, it is possible to calculate $ m_2 $ (measuring the values of the velocities of the particles before and after the collision is a standard procedure that can be carried out). 
A: Newton's Law are in addition to laws of force and mass.
Newton's law of mass, changes in mass are caused in proportion to changes in density and changes in amount of matter (this might be paraphrased too badly).
Force Laws (there are many, ones for gravity, ones for springs, etc.)
Newton's third law of motion constrains what force laws you consider (effectively you only use/consider force laws that conserve momentum).
Newton's second law of motion turns these force laws into predictions about motion, thus allowing the force laws to be tested, not just eliminated for violating conservation of momentum. This works because he postulates that we can test force laws by using calculus and then looking at the prediction from solutions to second order differential equations.
Newton's first law of motion then excludes certain solutions that the second law allowed. I'm not saying that historically Newton knew this, but it is possible (see Nonuniqueness in the solutions of Newton’s equation of motion by Abhishek Dhar Am. J. Phys. 61, 58 (1993); http://dx.doi.org/10.1119/1.17411 ) to have solutions to F=ma that violate Newton's first law. So adding the first law says to throw out those solutions.
In summary: the third law constrains the forces to consider, the second makes predictions so you can test the force laws, and the first constrains the (too many?) solutions that the second law allows. They all have a purpose, they all do something.
And you need to first have laws of mass and/or laws of forces before any of Newton's laws of motion mean anything.
