If a problem is in BPP, then there is an algorithm for it that is allowed to flip coins and make random decisions. It is guaranteed to run in polynomial time. On any given run of the algorithm, it has a probability of at most 1/4 of giving the wrong answer. That is true, whether the answer is YES or NO.
The choice of 1/4 in the definition is arbitrary. It can be any constant between 0 and 1/2 (exclusive) and the set BPP will be unchanged. The idea is that there is a small probability of error, but if the algorithm is run many times, the chance that the majority of the runs are wrong is exponentially small.
It is known that BPP=Co-BPP. It is an open question whether BPP is a subset of NP. It is an open question whether NP is a subset of BPP. If it is, then NP=RP. It is known that RP is a subset of BPP, and BPP is a subset of PP. It is not known whether those two are strict subsets.
The existence of certain strong Pseudorandom number generators imply that P=RP=BPP. This is now a widely believed hypothesis.
This class is defined for an ordinary Turing machine plus a source of randomness. The corresponding class for a quantum computer is BQP.