Should be used with the (group-theory) tag. For questions about braid groups: groups which arise as fundamental groups of configuration spaces and formalize the study of the everyday notion of a braid.

Algebraically a *braid group* $B_n$ is generated by elements $\sigma_1,\dotsc, \sigma_{n-1}$ subject to the relations
$$
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1} \;\text{ for }\; i \in \{1,2,\dotsc, n-2\}
\quad\text{ and }\quad
\sigma_i\sigma_j = \sigma_j\sigma_i \;\text{ for }\; |i-j|>2
\,.
$$
Intuitively, you think of $B_n$ as the group of braidings of $n$ strands, the $\sigma_i$ representing simple crossings between adjacent strands. Seeing illustrations of this is incredibly helpful in understanding braid groups, so please check out Wikipedia for a more thorough exposition.