The conformal bootstrap is a program for understanding conformal field theories in $d$ dimensions in terms of their $n$-point functions for (very) small $n$.
The case $d=2$ is rather special as the representation theory of the Virasoro group and the assumption of a central charge in the discrete series drastically restricts the possibilities and allows one to explicitly construct almost everything of interest. Because of the large symmetry group, everything can be done analytically. There are many books and lecture notes on CFT treating this in some detail; in the following, I do not consider this further.
In dimensions $d=3$ (relevant for statistical physics) and $d=4$ (relevant for relativistic quantum field theory), the conformal bootstrap remained dormant for many years since it was not clear what could replace the assumption of a central charge in the discrete series. Some important older papers are by Mack & Todorov, by Osborn & Petkou, by Nikolov, Stanev & Todorov; see also Fradkin & Palchik.
Recently, the conformal bootstrap lead to a flurry of new activity due to the discovery that at least the $d=3$ Ising CFT can be characterized as an extremal CFT according to several criteria. The assumption of extremality allows one to construct truncated optimization problems whose numerical solution gives values of unprecedented accuracy for the critical exponents of the Ising model. The central new idea was presented in a paper by El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin & Vichi. In my review of this paper, the theory underlying the principle is explained. (Almost everything else is just about computational techniques to create and numerically solve the associated optimization problems.) A recent quora discussion contains a long list of recent papers exploiting this idea.