There are different types of tyres you can have on your vehicle. What you choose depends on how you use your car, where you live and a myriad of other factors.
They are usually classified as:
Performance tyres are designed for faster cars or for people who prefer to drive harder than the average consumer. They typically put performance and grip ahead of longevity by using a softer rubber compound. Tread block design is normally biased towards outright grip rather than the ability to pump water out of the way on a wet road. The extreme example of performance tyres are "slicks" used in motor racing, so-called because they have no tread at all.
These tyres are what you'll typically find on every production car that comes out of a factory. They're designed to be a compromise between grip, performance, longevity, noise and wet-weather safety. For increased tyre life, they are made with a harder rubber compound, which sacrifices outright grip and cornering performance. For most drivers, this isn't a problem. The tread block design is normally a compromise between quiet running and water dispersion - the tyre should not be too noisy in normal use but should work fairly well in downpours and on wet roads. All-season tyres are what you should purchase when in doubt of what tyre you need.
Rather than use an even harder rubber compound than all-season tyres, wet weather tyres actually use a softer compound than performance tyres. The rubber needs to heat up quicker in cold or wet conditions and needs to have as much mechanical grip as possible. They'll normally also have a lot more siping to try to disperse water from the contact patch.
Winter (snow/ice) tyres
Winter tyres come at the other end of the spectrum to performance tyres. They're designed to work well in winter conditions with snow and ice on the roads. Winter tyres typically have larger, and thus noiser tread block patterns. In extreme climates, true snow tyres have tiny metal studs fabricated into the tread for biting into the snow and ice. The downside of this is that they are incredibly noisy on dry roads and wear out both the tyre and the road surface extremely quickly if driven in the dry.
All-terrain tyres are typically used on SUVs and light trucks. They are larger tyres with stiffer sidewalls and bigger tread block patterns. The larger tread block means the tyres are very noisy on normal roads but grip loose sand and dirt very well when you take the vehicle off-road. As well as the noise, the larger tread block pattern means less tyre surface in contact with the road. The rubber compound used in these tyres is normally middle-of-the-road - neither soft nor hard.
At the extreme end of the all-terrain tyre classification are mud tyres. These have massive, super-chunky tread blocks and really shouldn't ever be driven anywhere other than loose mud and dirt. The tread sometimes doesn't even come in blocks any more but looks more like paddles built in to the tyre carcass.