What is the decision version of independent set?

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 $\mathsf P$?

The definition of the decision version of independent set is the following:

Given as input a graph $G = (V,E)$ and an integer $k \in \mathbb{N}$, does there exist a set of pairwise non-adjacent vertices $I \subseteq V$ of size $|I| \geq k$?

The problem is polynomial time solvable if you consider $k$ to be constant, i.e., you can solve the problem in time $\mathcal{O}(n^k)$ but we usually take $k$ to be a part of the input. If you reduce an NP-complete problem to this problem, you will see that the $k$ is indeed unbounded.

StackExchange Computer Science Q#9363, answer score: 15