the gauge group of the standard model is SU(3)xSU(2)xU(1)

the SU(3) stands for special unitary matrices on a 3-dimensional vector space, which physically represents the symmetry between exchange of the 3 colors quarks can have. this it is sometimes called color SU(3) (to distinguish it from isospin SU(3))

the SU(2) stands for the symmetries present in the weak force, the quantum number being exchanged is called weak isospin.

the U(1) stands for hypercharge, another quantum number that particles can have.

a U(1) symmetry is the kind of symmetry that electromagnetism has, i.e. the electromagnetic field remains unchanged if you at a total derivative to the vector potential. the other gauge groups are similar, except they are nonabelian (i.e. given two matrices A and B, AB is not equal to BA unless they are 1x1 matrices, as in the case of U(1))

these are matrices of complex numbers. you can take the complex comjugate of each number in a matrix, and then exchange the rows with the columns. this new matrix is called the hermitian adjoint of the original. if a matrix is equal to its hermitian adjoint, then it is called a hermitian matrix. if the matrix is equal to the inverse of its hermitian adjoint, then it is a unitary matrix. if, in addition, the determinant of the matrix is 1, then it is called unimodular. the 'S' in SU(N) means unimodularity.