Why is the definition of mass and matter interlinked? In my textbook the definition of matter and mass are:


*

*Matter: Any thing that occupies space and has mass .

*Mass: The amount of matter  contained in a body.
While defining "matter" we refer to "mass", but the definition of "mass" refers back to "matter".
So isn't this wrong? What will be the right definitions?
 A: The word "matter" needs to be used with a great deal of care. In science, the word has definitely passed its use-by date. Some problems with the precise usage of the word are listed on the Wikipedia page for Matter and I quote:

...As such, there is no single universally agreed scientific meaning of the word "matter". Scientifically, the term "mass" is well-defined, but "matter" is not. 

Even mass is a word that needs to be used with great care. There are several, in-principle distinct meanings for it. Let's see whether we can untangle this mess.
For me and, I believe, most physicists, if one uses the word "matter" unqualified, it implies something with nonzero rest mass. Rest mass is the total energy of something that you measure when you are in an inertial motion frame that is at rest relative to the object in question (in SI units, you must divide the total rest energy by $c^2$ to get the rest mass). 
Inertial mass is the constant in Newton's second law. It is the "stubborness" of the object in question: it measures how quickly or slothfully something reacts when you "shove" it with an impulse. It is always measured from an inertial frame that is at rest relative to the body just before the shove. Whilst it is true that for Newtonian physics any inertial frame will do, in special relativity this frame must be at rest relative to the body and the impulse infinitessimal to get the correct inertial mass.
Gravitational mass is the "coupling strength" of an object to the gravitational field (if you're doing Newtonian Gravity) - how much gravitational force the object exerts on its surroundings and how much it itself is acted on by gravity. In the Einsteinian picture, it is a "source" strength of an object: it's part of what you plug into the so called "stress energy" tensor and so can be thought of as a "source" to calculate the warping of spacetime around the object. 
In turn, even the word "energy" which defines rest mass ultimately needs further qualification. In general relativity, it is the $0\,0$ component of the stress energy tensor: the "source" of spacetime curvature. Some people who study general relativity co-opt the word "matter" to mean "source of spacetime curvature", i.e. anything that begets a nonzero stress-energy tensor.
As far as we know, these three kinds of masses are the same. These equivalences are experimental facts (see for example the Eötvös experiment) and the gravitational / inertial equivalence is a beginning point for general relativity. A system with total energy at rest $m$ accelerates in response to a force $F$ at acceleration $F/m$. A system with total energy at rest $m$ experiences (in Newtonian gravity) a force $m\,g$ on it. The last equivalence, in particular, is tested constantly seeking tiny deviations from it in the hope of gleaning experimental data that would allow us to refine Einstein's theory of gravity. Look up, for example, the Nordtvelt effect. Or read up on the simple thought experiment by Paul Davies on quantum tunnelling in a gravitational field that suggests a deviation. The paper is cited in this Physics SE answer here
What are the modern usages? Physicists are most interested in states of the World where the quantum fields that make it up are in excited states. They have more precise names for these excitations, depending on which quantum field is involved: things like electron, photon, quark, muon .... and so forth. Unfortunately in general you need to specify exactly what it is that you're talking about. Also, mass by physicists is almost always taken to mean rest mass - this quantity is Lorentz invariant (independent of inertial observer). Sometimes you'll hear of relativistic mass - this notion has fallen into disfavor for teaching, but it refers to an object's increased inertial mass if it is moving relative to you. An impulse of $I$ will no longer beget a change of velocity $I/m_0$ when the body is initial moving relative to you (where $m_0$ is the rest mass) but rather the body behaves as though its inertial mass has increased to $\gamma\,m$, where $\gamma$ is the Lorentz factor.

As user Ruslan reminded me about the notion of relativistic mass:

Not quite $\gamma\,m$, rather it depends on angle between the force and velocity of the body. See discussion of longitudinal and transverse masses for details. 

A: These aren't good definitions. They do help students grasp the concept, but certain students, such as you, will find it silly that one definition depends on the other. Anyway, one definition depending on the other doesn't necessarily mean that a definition is flawed. In mathematics many functions are defined over itself, yet the definition is effective in that it either specifies a family of functions or, when given another defining component, fully define a specific function.
About the proper definitions, mass actually has three, but these are not very proper too: 
The inertial mass of an object is its resistance against acceleration by force.
The gravitational mass of an object is the object's tendency to be attracted by a gravitational field.
The relativistic mass is the mass of the total energy stored in a system, where mass is simply defined by the relation $E=mc^2$ to be a physical quantity directly proportional to energy, by a factor of $c^2 = 89875517873681760 m^2s^{-2}$. 
Going more hardcore into quantum mechanical definitions, mass becomes the property of a particle that measures/determines how strongly the particle interacts with the Higgs field.
The current best definition of matter is anything that is made up of leptons, fermions and bosons. These three are classes of subatomic particles called the elementary particles. Though everyday matter is only made up of half of these particles, it also encompasses other less "normal" matter such as strange matter. According to recent theories, these particles are merely field excitations and not rigid minuscule balls, so in the end, matter occupies no real space, but they need to stay a certain distance apart according to Pauli's exclusion principle, thus creating the abstraction that matter occupies space.(like ghosts, which do not take up space, but are prohibited from getting too close to each other) This Youtube video may help explain it: https://youtu.be/Fxeb3Pc4PA4 The video draws the conclusion that matter practically takes up space.
A: In my textbook the definition of matter and mass is as such: Matter: Any thing that occupies space and has mass. Mass: The amount of matter contained in a body. Why are the two definitions interlinked?
Because it's not a very good textbook. Take a look at Einstein's E=mc² paper, and note this: 
"The mass of a body is a measure of its energy-content".
It isn't a measure of its matter content, it's a measure of its energy content. Hence:
"If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²".
When you heat a body, you increase its mass. When you lift it up, you increase its mass. Whenever you do something that adds energy to that body, you increase its mass. Note though that when unqualified, mass means rest mass, so to keep things clear and simple, exclude those scenarios where the body is moving relative to you. 
Now if someone while defining matter asks me to define mass I cannot define mass using the terms of the earlier definition.
So define it in a better fashion, as resistance to change-in-motion for a body that's at rest relative to you. The opening paragraph of the Wikipedia article is more or less correct: 
"In physics, mass is a property of a physical body which determines the strength of its mutual gravitational attraction to other bodies, its resistance to being accelerated by a force, and in the theory of relativity gives the mass–energy content of a system."
There's "active gravitational mass" and "passive gravitational mass" and "inertial mass", all related to the energy-content. For a body such as a brick, these are the same as the rest mass, but not for a photon. A photon has no rest mass because it isn't at rest, but it does have a non-zero "active gravitational mass", "passive gravitational mass", and "inertial mass". Hence Compton scattering, and hence:
"If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies."
That's because things like inertial mass are really a measure of energy, and aren't quite the same thing as rest mass. Note though that when you trap a massless photon in a mirror-box, it's effectively at rest, and it increases the rest mass of the system. The box is harder to move when the photon is inside it, because the photon's resistance to change-of-motion doesn't go away when it's bouncing back and forth. Then when you open the box, it's a radiating body that loses mass. It's all fairly simple once it clicks. But sadly there's a lot of confusion around, and part of the reason for that is that the word "mass" is rather ambiguous. 
A: 
While defining "matter" we refer to "mass", but the definition of "mass" refers back to "matter". So isn't this wrong? What will be the right definitions?

Circular definitions are indeed undesirable, because they talk a lot but they tell little. The solution is to break the circle by removing the requirement to define the word and refer to experience instead. It ceases to be a mathematical definition, then, which is fine in physics.
In practice there is little problem with using the word matter. There is really no accepted definition, but that is quite alright; the word is similar to a fact, objective thing, which are not to be constructed by definition but accepted by  experience. In other words, you cannot learn what matter is from a textbook. You need to experience it.
A: Most textbooks define mass as the measure of amount of matter in an object which is a joke because that is a very elusive definition. Other books define inertial mass as resistance to change in motion in an object and gives a value equal to force/acceleration. Although the second definition seems rigorous, it is even more confusing because it presumes definition of force which is defined as force = mass * acceleration. A little contemplation will reveal that such definition of mass as force/acceleration is nothing but a circular definition.
As a student of physics I was thoroughly confused by such definitions. To make matters worse, pretty soon I learned theory of relativity only to discover that mass is relative. That means how much matter an object has depends on the observer's frame of reference - great!. This is unfortunately the state of affairs with current physics which entertains itself with exotic theories and concepts like leptons, hadrons, quarks, Higgs Boson but never addresses the fundamentals of physics.
After I passed out from school, I read many books on classical physics and nowhere I could find a definition of inertial mass to my satisfaction - until I came across the book Mechanics by L D Landau. His book gives correct definition of inertial mass which I will try to explain below.
In order to understand what inertial mass is, you have to first understand what an inertial frame of reference is. Again most text books do not define clearly what an inertial frame is. An inertial frame is a frame of reference in which space is homogeneous and isotropic, and time is homogeneous. Space is homogeneous means all points in space are equivalent i.e. laws of physics are same in every point of the inertial frame. Stated alternatively, a translation of an inertial co-ordinate system will not alter the laws of physics for the observers in that system. Space is isotropic means all directions in space are equivalent or a rotation of the co-ordinate system will not alter the laws of physics. Time is homogeneous means all points of time are equivalent. Stated alternatively, that laws of physics will not change with time.  Now that you understood what an inertial frame is, you are ready to understand what an inertial mass is. 
Classical mechanics is based on the fundamental principle called principles of least action or Hamiltonian principle. This principle states that any mechanical system with with N degrees of freedom can be represented mathematically by a Lagrangian function of the form L(q1, q2 ... qN, t), where q1, q2 ... qN are variables called the co-ordinates of the system. At a given point in time, the system is completely defined by the values of the coordinates q1, q2, q3 .... qN.  When the system moves from state S1 to S2, the system will follow the path for which the time integral of the Lagrangian is minimum.
Now consider a universe in which there nothing except an inertial frame and a particle moving in that frame. Since space is homogeneous and isotropic, time is homogeneous, it can be shown that such a particle will move at a constant speed with respect to the inertial frame and the Lagrangian of the particle will have the form 1/2mV^2 where V is the speed of the particle and m is a constant. The constant m is called the inertial mass of the particle.
