What is an inertial frame? Throughout my life I've been told that an inertial frame is one that is not accelerating and I was satisfied with that. Well up to this day, until I asked: accelerating with respect to what ? Now this seems as a flawed definition to me.
Another definition states that an inertial frame is one in which a body remains at rest or moves at constant velocity unless acted upon by forces. This definition seems to be flawed too, don't we usually tell whether or not a force is acting on a body by measuring its acceleration ? How can we switch to doing the opposite so quickly ?
Wikipedia defines an inertial frame as:

In classical physics and special relativity, an inertial frame of reference (also inertial reference frame or inertial frame, Galilean reference frame or inertial space) is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner.[1] The physics of a system in an inertial frame have no causes external to the system.[2]

Well this looks too complicated for me, can any one help explaining it in simpler terms please ? How can time and space be "homogeneous", "isotropic", and "time-independent" ?
Finally, can we define an inertial frame as the one in which Newton's three laws of mechanics hold? 
 A: Your question ultimately boils down to "Accelerating with respect to what?"
But the first factor to account for is that accelerations are not relative.  Given two astronauts drifting in space with a constant velocity (relate to one another), there is no way to identify which one is moving and which is stationary - the answer is intrinsically a matter of perspective.  One way of expressing it is that if they were each enclosed in a capsule, there is no possible experiment they could perform that would tell them whether their velocity was zero, or non-zero.
But two astronauts accelerating relative to one another is not a matter of perspective.  They both agree that the one wearing the rocket-pack is accelerating and the one without a rocket-pack is not accelerating.  Accelerations are caused by forces and forces are caused by an interaction with an outside object (the rocket fuel, in this case).  Either that interaction exists, or it doesn't.  If they were enclosed in capsules (that have thrusters on the capsules), there are lots of experiments that would tell them whether their acceleration was zero or non-zero.  For instance, release a ball and see if it hits the floor or floats.
A: The Wikipedia definition doesn't say "Space is homogeneous, isotropic, and time-independent." It says that you can sometimes describe space in a homogeneous, isotropic, and time-independent way, and if you can do that, you are using an inertial reference frame. You can describe the same region of space using different reference frames, some inertial and some non-inertial. The descriptions will be different in the inertial and non-inertial frames, but both are describing the same space.
To explain what the three terms mean in simpler words, imagine you have set up a physics lab on a spaceship with no windows, (so you can't look at anything outside the ship to see how the ship is moving) and you want to know if a reference frame fixed to the ship is inertial.
"Homogeneous" means an experiment always gives the same results when you do it in different places in the lab.
"Isotropic" means an experiment always gives the same results if you change the orientation (as a simple example, suppose you rotate the apparatus 90 degrees on your lab bench, and do the experiment again).
"Time-independent" means an experiment always gives the same results if you repeat it at different times.
If the frame fixed to the ship is not inertial (for example, the ship is accelerating or rotating), you can invent examples where those criteria would not be satisfied.
A: To describe the world one needs a frame of reference. But there are so many of them! For example I can choose a system of reference attached to my head. When I turn my head a lot of interesting things happen. For example you suddenly "jump" several kilometers or thousands kilometers, depending on how far you are from me. With no obvious reason: nobody was pushing or pulling you. Stars (including the Sun) also jumped, moving faster than light by the way.
It is still possible to use this system of reference to describe the world. And it is even possible to create more weird ones. But the laws of nature would be very strange when written down in these systems.
It turned out that our world is not that much complicated. There are very good, convenient, "natural" systems of reference, such that the laws of nature are more or less simple when written down in these systems. For example, if nobody touches some body this body does not accelerate. And if body accelerates it's possible to find out why, what other bodies influence it.
These good and convenient frames of reference are called inertial. The Newton's First Law states that such frames of reference exist in our Universe.
