Are there nontrivial vector spaces with finitely many elements? I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space?

Any vector space over an infinite field (like $\Bbb Q$ $\Bbb R$ or $\Bbb C$) will either be the space $\{0\}$, or will have infinitely many elements.  In particular, for any non-zero vector $x$ and any field-elements $k_1,k_2$, the elements $k_1x, k_2 x$ are only equal when $k_1 = k_2$.  So, there are infinitely many vectors of the form $k\,x$ as long as our vector space is non-zero over an infinite field.
There are, however, finite fields.  For example, we can define multiplication and addition on $\Bbb F_2 = \{0,1\}$ modulo $2$. The vector space $\Bbb F_2^n$ is finite for any $n$ (and has, in particular, $2^n$ elements).  For example, we have
$$
\Bbb F_2^2 = \{(0,0),(0,1),(1,0),(1,1)\}
$$
