In other words, there is an algorithm for a quantum computer that is guaranteed to run in polynomial time. On any given run of the algorithm, it has a probability of at most 1/4 that it will give the wrong answer. That is true, whether the answer is YES or NO.
The choice of 1/4 in the definition is arbitrary. Changing the constant to any real number k such that 0 < k < 1/2 does not change the set BQP. The idea is that there is a small probability of error, but running the algorithm many times produces an exponentially-small chance that the majority of the runs are wrong.
The number of qubits in the computer is allowed to be a function of the instance size. For example, algorithms are known for factoring an n-bit integer using just over 2n qubits.
Quantum computers have gained widespread interest because some problems of practical interest are known to be in BQP, but suspected to be outside P. Currently, only three such problems are known: