Let $e_{11}, e_{12}, e_{21}, e_{22}$ denote the standard basis of the real vector space $M_{2\times 2}({\mathbb{R}})$ of $2\times 2$ matrices with real entries. That is to say, the matrix $e_{ij}$ has a $1$ in the $(i,j)$th entry and a $0$ elsewhere.
Now viewing $M_{2\times 2}({\mathbb{R}})$ as an algebra over the real numbers, what is the kernel of the algebra epimorphism $\phi: {\mathbb{R}}[e_{11},e_{12},e_{21},e_{22}] \to M_{2\times 2}({\mathbb{R}})$ which sends each $e_{ij}$ to itself?
More precisely, can a finite collection $f_1(e_{11},e_{12},e_{21},e_{22}),\ldots, f_n(e_{11},e_{12},e_{21},e_{22})$ of polynomials in the variables $e_{11},e_{12},e_{21},e_{22}$ be explicitly given such that $\ker \phi$ is the subalgebra of ${\mathbb{R}}[e_{11},e_{12},e_{21},e_{22}]$ generated by $f_1,\ldots, f_n$?
Over any commutative ring $k$, you can present $M_n(k)$ as a $k$-algebra by generators $e_{ij}, 1 \le i, j \le n$ and relations
$$e_{ij} e_{jk} = e_{ik}, 1 \le i, j, k \le n$$
and all other products of the generators are zero. Equivalently, it's the category algebra over $k$ of the category consisting of $n$ objects $\{ 1, 2, \dots n \}$ each of which is uniquely isomorphic via a unique isomorphism $e_{ij} : i \sim j$.
In general, given a finite-dimensional algebra $A$ and a basis $e_i$ of it, you can write down a presentation just by writing down all relations $e_i e_j = \sum c_{ijk} e_k$.