What is the difference between a measurement and any other interaction in quantum mechanics? We've learned that the wave function of a particle collapses when we measure a particle's location. If it is found, it becomes more probable to find it a again in the same area, and if not the probability to finding it in the place that was checked decreases dramatically.
My question is about the definition of measurement. What makes a measurement different from any other interaction between two particles (gravity and EM fields for example)?
In reality, almost every particle interacts with any other particle, so shouldn't there be constant collapse of the wave function all the time? If this happens we're right back in classical mechanics, aren't we?
 A: Much of how you answer this question comes down to your view of the wavefunction or state. If you think that the quantum state is a state of reality (that is, an ontic state), then you must either reproduce the predictions of orthodox (Copenhagen) QM without the measurement postulate or you must explain why nature provides two forms of evolution. The former view is basically the Many Worlds Interpretation, which I feel a great degree of attraction to, as it postulates only unitary evolution, and explains measurement as being an emergent, rather than fundamental, effect.
On the other hand, if you hold that the wavefunction is a state of knowledge (epistemic) about some other underlying ontic state, then measurement collapse represents not a true evolution, but a discontinuous change in your knowledge about a system. Alternative formulations of quantum mechanics, such as Bohmian mechanics, explain this in a mathematically rigorous way, but that some find unsatisfying.
Each of these approaches (and the many more I didn't mention) suggests where to look for the next physical theory, and so the question should eventually be experimentally decidable. For now, though, we must rely on mathematics, physical intuition and rational argument.
A: Much has been covered in these answers, but one aspect has been left out.
The actual physics going on in any measurement process includes amplification.  Feynman thought this was significant.  Here is a perhaps little-known quotation of his:

We and our measuring instruments are part of nature and so are, in principle, described by an amplitude function [the wave function] satisfying a deterministic equation [Schrodinger's equation]. Why can we only predict the probability that a given experiment will lead to a definite result? From what does the uncertainty arise? Almost without a doubt it arises from the need to amplify the effects of single atomic events to such a level that they may be readily observed by large systems.
\dots In what way is only the probability of a future event accessible to us, whereas the certainty of a past event can often apparently be asserted? \dots Obviously, we are again involved in the consequences of the large size of ouselves and of our measuring equipment. The usual separation of observer and observed which is now needed in analyzing measurements in quantum mechanics should not really be necessary, or at least should be even more thoroughly analyzed. What seems to be needed is the statistical mechanics of amplifying apparatus.

R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, New York, 1965, p. 22.
This is quoted and discussed in my The Axiomatisation of Physics, see
http://www.mast.queensu.ca/~jjohnson/HilbertSixth.pdf
and
http://arxiv.org/abs/0705.2554
A: It is the case that all measurements proceed via the exploitation of the natural interactions that we understand theoretically. But once the measurement is completed and the result in hand, the QM analysis of the subsequent evolution of just those systems that yielded that particular result can no longer employ the original state function (which allows for all the different possible results), but must then employ just that part of the original state function that corresponds to the particular result. This 'sudden' change in the state function used is called state function collapse. Many physicists regard this change as corresponding to nothing more than the change in the experimenter's knowledge once the result is in hand. This is the epistemological interpretation of the state function. But many regard the change as also reflecting a genuine physical change in the state of those systems that came through the measurement yielding the particular result. This is the ontological interpretation of the state function and it has many variations. Still many others hold an ontological interpretation of the state function while denying that collapse happens at all. 
These latter views, which also have many versions, have given rise to various interpretations of and/or alternatives to QM that go by names such as Pilot wave, deBroglie-Bohm, Modal interpretations, Relative state, Many Worlds, Many Minds, Consistent Histories, Decoherence theoretic, Information theoretic etc. Collectively these are all called NO-Collapse theories.
The champions of real, physical collapse have also been at work creating alternative theories of their own that replace the collapse postulate by evolutions that generate collapse dynamically. These theories go by the names of their authors, Ghirardi-Rimini-Weber-Pearl, Karolahazy, Penrose, Gisin, Percival, etc. Collectively these are the Collapse theories.
The difficulty in deciding among these many and still proliferating alternatives is due to the incredible success of standard QM. All the alternatives must, at least, reproduce the corroborated results of QM while possibly allowing for deviations in, as yet, untested waters. Some of them offer no deviations from QM, whatsoever! So deciding between them 
and QM must be a matter of philosophy or aesthetics. In any case, the days of the hegemony of the Copenhagen interpretation, if they ever really existed, are gone forever.      
A: What you describe in your question is the "Copenhagen interpretation" of quantum mechanics.  There are more nuanced views of this nowadays that don't treat "measurements" quite so asymmetrically, see e.g. sources that talk about decoherence.
I recommend watching the classic lecture "Quantum Mechanics in your face" by Sidney Coleman for a nice take on this sort of thing.  
A: Maybe this is oversimplifying things but:

*

*the interaction between two phenomena occurs when they mutually change their properties: $p_1$ changes $p_2$ and $p_2$ changes $p_1$. And there is no possibility that one phenomenon affects the other one without being changed itself.


*A measurement is a kind of interaction where some information about the value of a property of, say $p_1$, can be inferred from the changes in $p_2$'s properties after they have interacted. The value obtained is always subject to a non-zero degree of uncertainty.
A: Interactions merely involve a correlation developing.  For example, if an electron is put through a Stern-Gerlach apparatus, a correlation develops between the distance travelled in the x direction and the distance deviated in the y direction.   It is reversible. The measurement which occurs when the particle hits the photographic plate is irreversible.   It is associated with irreversible dissipation, i.e. entropy generation.
This approximation can itself be dissected further, but it gets very tricky.
A really good (1983) book is by Wheeler and Zurek, "The quantum theory of measurement" available as a djvu file at http://www.4shared.com/get/vw66Qp70/Wheeler_JA_Zurek_WH_eds_Quan.html  (8 MB, wait 30 sec for the download).    [Now if I can only figure out how to work a reader for a Mac ...]
A: This is a question that philosophers of physics try to answer now, not physicists (even if most of the times the border is not really sharp). So if you are looking for a more detailed discussion (and ressources), you should have a look at this article of the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/qt-measurement/
A: The measurement happens at the end of time for our universe as a collapse in the form of post-selection as in the two-state formalism. Read the papers by Aharonov and Vaidman for more details. There is a theorem in quantum mechanics to the effect that we can always push the measurement collapse into the future without any observable consequences.
A: The short answer is that measurement and interactions are two different animals in Quantum Mechanics. In reality measurements are performed using one of the fundamental interactions (usually EM), but this does not enter the framework of QM.
The long answer is that you will not receive a satisfying answer to your question. First, because physicists don't know the answer, and second, because physicists don't care.
Physics is concerned with understanding nature, in so far as making predictions regarding measurements. If we have a theory of what happens between measurements (things like Lagrangians and forces) and a theory of measurements (a postulate in Quantum Mechanics that wave functions collapse, plus the probabilistic interpretation of the abs.square of the wave function), and this framework works to the desired accuracy, then the philosophical implications of trying to unite the two are of no interest to physicists, unless they bring about a deeper understanding of nature, in so far as making more precise or more general predictions regarding measurements.
In practice, the line of questioning you pose has been investigated ever since the advent of Quantum Mechanics, but to my knowledge, nothing ever came of it regarding unification of forces and measurements ("don't know"), so the mainstream has lost interest a long time ago ("don't care"). (As an interesting side-note, one important result to come out of related type of inquiry is the Bell inequality.)
Sorry if this answer seems negative. To quote David Mermin [corrected] regarding the philosophical issues regarding Quantum Mechanics, the pragmatic thing to do is to "shut up and calculate!"
A: It seems that this question is regarding Copenhagen interpretation (and other related interpretations with collapse and one observer) solely, because it uses the term "measurement" which has no special position in other interpretations. 
So assuming this is about the Copenhagen interpretation, yes, the measurement is different from any other interaction. The difference is in that the quantum system interacts with the observer, a person who has special physical properties of having ability to trigger the wave function collapse. There is only one such person and the QM provides the theoretical possibility to unambiguously determine who is it based on his special abilities of interacting with matter.
This constitutes the main problem of Copenhagen interpretation, and the reason why other interpretations were proposed (Relational, MWI) which are observer-agnostic, do not include special, chosen personalities and symmetric across all people. This does not mean however that still there should not be a person apparently having special properties at least in the observable universe. 
