The Single-coil pickup is probably the most common pickup type. They are renowned for their ‘open and bright’ sound, which can vary from ‘pristine and jangley’ to ‘full and warm’, dependant on where the pickup is placed and whether or not it is used in conjunction with other pickups on the guitar. One downside of Single-coil pickups is that they’re prone to hum/background interference; especially if you’re using effects, in particular, distortions and overdrives.
A Humbucker is essentially 2 Single-coil pickups, wired together with opposite magnetic polarities and coil windings. They are the pickup of choice for rock and metal, and really anything where distortion is being used (due to their hum-cancelling properties). Humbuckers are a fair bit bassier than Single-coils and they have a much higher output making them perfect for high-gain applications.
Active vs. Passive
Active pickups have their own preamps (powered by a 9v battery) allowing for more tone shaping through the use of cut/boost circuits for certain frequencies. Active pickups generally are cleaner and clearer than passive but are also more compressed and have a much higher output giving them less dynamic range. Passive pickups are generally warmer in sound and mostly have a lower output giving them more headroom; meaning they respond a lot more to picking dynamics.
Ceramic vs. Alnico
Ceramic magnets are very clean and sterile sounding with relatively high output making them the perfect choice for active pickups or passive pickups looking to imitate the characteristics of an active pickup. Alnico (aluminium (Al), nickel (Ni) and cobalt (Co)) pickups are generally warmer sounding and are often referred to as being ‘more musical’ as the Alnico magnets have a lot of their own tonality rather than being super clean like their Ceramic counterparts.