PNG stores exactly the same pixels as a BMP, but compressed without loss, so the picture is identical and the file is usually a fraction of the size. Screenshots, diagrams, icons and other flat graphics shrink the most, because PNG's compression is built for large areas of the same colour.
The converter opens each bitmap with your browser's image decoder and writes a PNG with every pixel unchanged. Palette (1, 4 and 8-bit) and true-colour (24 and 32-bit) BMPs are supported, and transparency in 32-bit BMPs is carried over to the PNG.
Photos also convert perfectly, but a lossless PNG of a photo stays fairly large; if you need a small photo file, BMP to JPG is the better fit. There are no settings to get wrong: PNG has no quality level to choose.