The braid group on $n$ strands, denoted $B_n$, has a geometric/topological definition as the group whose elements are equivalence classes of $n$-braids (under ambient isotopy), and whose group operation is composition of braids. See Wikipedia for more details.
The standard presentation of the braid group $B_n$ is $$\langle \sigma_1,\ldots, \sigma_{n-1}\mid\sigma_i \sigma_{i+1}, \sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1} \text{ for }1\leq i\leq n-1, \text{ and } \sigma_i\sigma_j=\sigma_j\sigma_i \text{ for } |i-j|\geq 2\rangle.$$
Geometrically we can verify that these relations indeed hold, and so $B_n$ is a homomorphic image of the group with this presentation. But how do we know if these are all the possible relations there can be?