Bits are the smallest units of data that computers can use. A single bit is a binary digit. It can be a 1 or a 0. Bits are the medium computers use for any kind of information.
Bit depth is the number of bits used to encode a sound file. Sound files have "noise floors" and "noise ceilings," which limit how loud or quiet the sound can be. Greater bit depth pushes the floor down and the ceiling up.
A sound file's bitrate shows how much data must be processed per second to play the file: greater bit depth leads to greater bit rate. Sound files with high bit rates will take up more space but sound more clear.
When you use a program to convert a file's bit depth, you're changing how much data is allotted to the file. A lower bit depth can result in lower quality and lower bitrate, but the file will take up more space. The opposite applies to higher bit depth.