BPP is the complexity class for decision problems that a randomized algorithm can solve in polynomial time, in the sense that on every input (worst-case), the algorithm is right with at least 2/3 probability (over its own randomness).

One example is deciding whether a given number is prime. For a long time, we knew randomized algorithms like the Miller-Rabin test that are correct with high probability. That is, we knew PRIMES is in BPP. But we didn't know a deterministic algorithm. In 2006, a deterministic algorithm was discovered, proving that PRIMES is in P.

One of the central open problems in Computer Science is whether BPP=P. That is, for any problem we can solve with randomness (in the above sense), can we solve it without randomness? Among many other contributions, Wigderson's work has made a ton of progress toward this problem.

The general idea is to try to "derandomize" any randomized algorithm that runs in polynomial time and is correct with 2/3 probability. We will feed it inputs from a pseudorandom generator (instead of true random numbers). If the PRG is really really good, then we can get a deterministic algorithm by running the original algorithm along with the PRG.

Now, if we can prove certain problems are hard, then we can also prove that such really really good PRGs exist. This is basically because telling apart the PRG from true randomness would be a computationally hard problem.