Overdubbing anything simply means recording certain tracks at a later time than when the initial basic tracks (usually rhythm tracks in pop music) were recorded.
Overdubbing in and of itself has no particular sound.
Perhaps you are confusing this with the technique called "doubling"? Doubling does have a characteristic "sound". It is often done with lead vocals in certain parts (or all) of a song. The vocalist simply resings the lead (while listening to the original lead track) and tries to reproduce it as closely as possible. The two are then blended together to create a "doubled" sound. Sometimes they are blended at different volumes, or different processing and fx are used on the doubled track, depending on the sound you are looking for. In general, it fattens up the sound, and sometimes can camouflage questionable intonation.
You could also triple or quadruple to get an even more pronounced effect.
This technique is not limited to lead vocals - it is often used on string parts, guitars, BU Vox, and others.