The number of times wire is wrapped around the bobbin determines the intensity of the magnet field. This creates different types of pickups made for certain genres of music. A more relaxed style, which would want a softer tone for a classic sound, would have under 10,000 wraps of wire. However, metal guitarists would prefer upwards of 18,000 wraps of wire because it creates super high gain used in metal. By adjusting the amount of wire, you can adjust for what sound you want.
Different types of magnets are used in the pickups. Alnico and ceramic are the most common with a Hybrid sometimes used. Alnico magnets are divided into three types. The II, III and V are different alnico magnets. The II is a middle ground between the III, which offers a softer tone having the least pull, and the V, having the most aggressive pull. Ceramic magnets has a very thick tone which creates a full sound. The Hybrid is a signature pickup with an Alnico V and a Ceramic magnet.
Another big difference is made when using either a single coil or humbucker pickup. A single coil is, as the name implies, a pickup with one spool of wire. They have a relative clean sound, producing a trebly output with very little bass. The most common guitar manufacturer that uses these is Fender. Many country guitarist love the "squeak" of the Fender Stratocaster and Telecaster obtained by using single coil pickups. In contrast, a humbucker has two spools of wire. They have a very hot output, with much bass and very little treble. It's easy to get a crunchy sound from humbuckers even when your amp is set for a clean channel. Gibson is the big name mainly associated with using humbuckers. Generally, people use these for rock style music. Both are great, but neither could do the job for everything. Ultimately, it's about your preference which is most influenced by your fellow peers and guitar icons.
Pickups can be either passive or active, passive being more common and cheaper. Passive pickups receive their power through the amp via the input cable. When the amp is turned on, the electricity flows through the pickup to create the magnetic field. However, active pickups require a battery to power the pickup. They have a lower output so it uses the battery as a preamp to boost the guitar's output signal. When you plug the cable into the guitar, the battery automatically turns on the preamp, pickups and all. To shut the battery off, you simply unplug the cable from your guitar's input jack.
Shielding is the final step to complete the pickup. You shield it to prevent unwanted buzzing produced by UV Rays hitting the magnetic field. Generally, wax is used for cheap ones and the more expensive ones have either a spray or some type of tin foil like paper. Without this, they pickup more than just the string's vibrations.
Pickups make a world of a difference in the world of guitar electronics. They are the single most influencing thing on the sound and tone of your guitar. In order to obtain your desired sound, you have to choose the correct style of pickup. With many variables, you must do some research to get an idea as to what you would want. Always, always play a guitar with the pickups you want before buying them.