A brane is a feature of more advanced string theory, its a piece of spacetime of some number of dimensions (almost any number). The word brane comes from "membrane", and the idea is "Like that but in several dimensions" Originally branes and strings were more or less on an equal footing, and branes were where strings stuck their end points (if I may be so over colorful). In later work it's branes on branes, and crossed branes, and all sorts of variations.
LQG stands for Loop Quantum Gravity. About 20 years ago it was found that you can do interesting things with "Wilson loops" that were sort of physics around a loop in space. Then it was found they didn't have to be loops (but the name stuck) and they started using networks. The idea is to use these networks to build quantum gravity of spacetime. The traditional way of quantizing gravity, they way they had quantized the other forces, didn't work.
Now both advanced string physics, with branes, and Loop Quantum Gravity, without loops, are candidates for quantizing gravity. May the best theory win. And by "best" I mean the one that actually gets the job completely done. If they both do, even better.