B3LYP is a hybrid functional developed in the late 80s. It turns out that DFT and Hartree-Fock based methods are basically trying to do the same thing - recover electron correlation. However, they have different difficulties, Hartree-Fock methods exactly treat exchange correlation but have difficulties recovering dynamic electron correlation while DFT has an exact form for dynamic electron correlation but since DFT is not quantum mechanical, it must approximate exchange correlation.
B3 is Becke's 3 paramater exchange correlation functional which uses 3 parameters to mix in the exact Hartree-Fock exchange correlation and LYP is the Lee Yang and Parr correlation functional that recovers dynamic electron correlation (I think they used a fit to ab initio calculations on a He atom). B3LYP is popular for many reasons. It was one of the first DFT methods that was a significant improvement over Hartree-Fock. B3LYP is generally faster than most Post Hartree-Fock techniques and usually yields comparable results. It is also fairly robust for a DFT method. On a more fundamental level, it is not as heavily parameterized as other hybrid functionals, having only 3 where as some have up to 26. Becke's original paper is one of the most cited papers (I think it's number 8) of all time, so B3LYP is well established in the literature and people are less likely to complain about your choice versus a newer functional. With that said, you should always look in the literature before you start doing DFT calculations on a system to see if other functionals have been shown to do better.