Choosing Pickups
Single Coils vs. Humbuckers
One of the largest issues involved in magnetic guitar pickups is that they are prone to picking up interference, such as from flourescent lights or radio signals. This produces hum and noise. Single coil pickups use one coil, whereas humbuckers use two coils wound in opposite directions. Humbuckers pick up the interference, but with opposite voltages in each coil, causing the interference to cancel out. This is why they are called humbuckers - they have less noise and hum than single coils. This is the main difference between the two.
Humbuckers also have more output than single-coils, which can make them more easily push an amp into overdrive/distortion. In the modern world, this isn't much of a feature - most amps now feature powerful pre-amp drive. Or you can resort to a fairly cheeap boost pedal to make the signal hotter. Humbuckers main benefit is their higher signal-to-noise ratio. Distortion tends to add noise, so it's helpful to start with a rather clean signal, or things can quickly become a noisy mess.
Yet, single-coil and humbucker pickups have different tone. This is mostly due to their impedance. Pickups have a resonant peak, a frequency where they are most responsive and have more output. Beyond the resonant peak, pickups get worse and worse at reproducing those frequencies. Single-coils have a higher resonant peak, and thus have a certain "shimmer" to their tone. They are described as glass-like. Humbuckers, in contrast, sound a bit nasal.
Parallel-wired Humbuckers
The above descriptions refer to humbuckers wired in serial, which is the most common way to find them. The ending terminal of one coil is wired directly to the beginning terminal of the other coil. However, many manufacturers provide "4-wire" pickups. Each coil is wired independently, and the customer is given the two endpoints for each coil (hence 4-wire). You are free to wire each coil as you like.
One way to wire them is in parallel. This makes them sound similar to single coils - they have the lower output and higher resonant peak. But at the same time, they still cancel hum/noise from interference. Because they are capturing the string at two "points" as opposed to the smaller one "point" of a traditional single coil, there may be some high frequency cancelation (due to a comb filter effect), so they won't sound exactly like a single coil. But it is rather close.
The point here is just because it's a humbucker doesn't mean it has to sound nasal.