[This is an easy question, but it doesn't feel like a homework question, so I will answer it. I have made the post community wiki to protect myself from unwanted votes, both upwards and downwards.]

For a positive integer $n$, consider the ring $M_n(k)$ of $n \times n$ matrices with $k$-coefficients for $n \geq 1$.

If $k$ is finite, then $M_n(k)$ is finite, so obviously there are only finitely many similarity classes.

If $k$ is infinite, then since the determinant map $M_n(k) \rightarrow k$ is surjective and the determinant is a similarity invariant, there are infinitely many similarity classes.

One may ask a more precise question: what is the cardinality $S(n,k)$ of the set of
similarity classes of $n \times n$ matrices with coefficients in $k$?

When $k$ is infinite, it follows easily from the above that $S(n,k) = \# k$.

On the other hand, when $k \cong \mathbb{F}_q$ is finite, it is a nice linear algebra exercise to give an explicit formula for $S(n,k)$ in terms of $q$ and $n$. It might (or might not) be appropriate to discuss how to derive such a formula here.