There are basically two types of pickups for electric guitars: single coils and humbuckers; and they sound completely different. Single coils tend to be softer and brighter (and electrical interference will cause them to hum); humbuckers tend to be louder and have much stronger midrange and bass response (and they don’t hum). In addition, single coils tend to have better clarity than humbuckers when played clean, but humbuckers tend to work better with overdrive or distortion (because they are more powerful). Single coils also tend to sound better in the neck position, and humbuckers tend to sound better in the bridge position (again because of the midrange response and the additional power).
There are a number of pickups marketed as single coils that don’t hum, including Fender’s Vintage Noiseless pickups and Lace Sensor’s “Holy Grail” pickups. For the most part, those types of pickups are actually tiny, bright sounding humbuckers http://itsnews.co.uk/ . They are made to look like single coils by stacking the two coils on top of each other, instead of laying them side by side. No matter what anyone tells you the only thing that really sounds like a single coil pickup is a single coil pickup.
I think a better way to solve the hum issue is to get a reverse wound reverse polarity (rwrp) middle pickup (Fender Custom Shop Fat 50’s have a rwrp middle pickup). That way, if you have a Stratocaster, for example, you will have single coil tone in positions 1, 3 and 5, but you will have no hum in positions 2 and 4. Alternatively, if you have a Les Paul, you could get humbuckers that allow you to split the coils, so that you can convert each humbucker to a single coil with the flip of a switch (Seymour Duncan JB Model humbuckers have four conductor leads, so you can use them with a coil splitting switch). Either way, you can get the best of both worlds.
Among single coil pickups and humbuckers, there are many variations in how they are constructed and how they sound. Basically, a pickup is a row of magnets wrapped in copper wire. So changes in the magnets and the wire affect the sound. Alnico V magnets are commonly used in single coil pickups, like Fender’s Texas Special pickups for Stratocasters and Telecasters; they are stronger magnets and have a sharper sound. Alnico II magnets are more common in humbuckers, like Gibson’s Classic ’57 pickups; they are softer magnets and they have a smoother tone.
As for the copper wire, “overwound” pickups tend to sound louder and have more midrange and bass; pickups with less windings tend to sound softer and brighter. One of the reasons humbuckers sound the way they do is because it takes more wire to wrap the two coils. The thickness of the wiring and the type of insulation that is used are additional factors that affect the sound (e.g. Fender’s early Strat pickups had Formvar insulation instead of enamel; insulating them that way gave them a clearer tone). Today most humbuckers are also wax potted so they won’t squeal at high gain, but the wax potting hurts the clarity a little too (Gibson’s modern Burstbucker pickups and Seymour Duncan’s Seth Lovers attempt to reproduce the clearer tone of early humbuckers by eliminating the wax potting).
Another thing to consider with single coils is how the construction will affect the way the pickup responds to electrical interference. You may love the way a big, fat single coil like a Gibson p90 sounds, but you may also find the extra wiring that makes the pickup sound so good makes it hum louder too. So there is a trade off if you like that sound (more wire = louder, fatter sound = more hum).