Question: This is a question that has been posted at many different forums, I thought maybe someone here would have a better or more conceptual answer than I have seen before:

Why do physicists care about representations of Lie groups? For myself, when I think about a representation that means there is some sort of group acting on a vector space, what is the vector space that this Lie group is acting on? 

Or is it that certain things have to be invariant under a group action?
maybe this is a dumb question, but i thought it might be a good start...

To clarify, I am specifically thinking of the symmetry groups that people think about in relation to the standard model. I do not care why it might be a certain group, but more how we see the group acting, what is it acting on? etc.

Answer: 
The vector space that is being acted on typically is a Hilbert space of states in quantum mechanics; very roughly, there's a basis of this vector space which is in one-to-one correspondence with the set of possibilities for a physical system. The simplest example to try to get your head around is that of the spin 1/2 particle (2 dim representation of SU(2)), which is explained in any introductory quantum mechanics book.
See the [Wigner theorem](http://en.wikipedia.org/wiki/Wigner_theorem), it explain rigorously the relationship between a group of symmetries and states of a physical particle.