The main difference between these two options is that one of them has one coil of wire that goes wrapped around one or more magnets, this being the single-coil, while double-coil units come with two wires wrapped around, based on the same principle.
You should also keep in mind that double-coil pickups usually have more output, which means they can render more pronounced sounds. Moreover, they don’t pick up “hums” either, given that the two wires work together and cancel out any interference.
In fact, this is one of the reasons for which they are a very popular choice among guitarists. In terms of the way they are made, double-coil picks can be set in series or in parallel, each option having a slight impact on the sound as well. However, the choice here depends on the user’s preferences as well, so you should try more alternatives before picking your favorite.