I always read that finding an independent set of size $k$ in a graph is $\mathsf{NP}$-complete. However, this only requires looking for all combinations of $k$ vertices and this is a polynomial procedure of order $k$.
I know that we can reduce directly SAT to independent set, with $k$ the number of clauses.
The problem is that I can't grasp correctly, as in 3-COLORING or 3-SAT, the required format to study the complexity of INDEPENDENT SET.
What is the decision version of independent set? And why isn't $k$-independent set in $\mathsf P$?