I've always heard of such statements as $SU$ group describing our world even before I seriously learned physics.   When learning about spin, someone spoke of $SU(2)$ but did not seriously explain. When learning isospin and hypercharge for particles made of $u, d, s,$ someone spoke of $SU(3)$ but did not give detailed explanation. When talking about the discovery of J/ψ, teacher (a experimentalist) say, well now with the charm quark, we need $SU(4)$.   I am crazy, $SU$ is just special unitary matrix group, what are they talking about when relating these matrix with spin or particles? I hope someone could give me a detailed explanation about how these group is related to real world substances or recommend a book.
I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?  I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$?