There are a lot of different tyres around, but I'll have a go at explaining in general terms. If I've got anything wrong, someone will correct me.
Tyres in general will vary by; diameter, width, rolling resistance, weight, rubber compound (grip), puncture resistance and tread pattern. What you use the bike for affects which of those characteristics will be more important. If you are commuting, you may favour puncture resistance, if racing then rolling resistance and weight become more important. Most popular brands seem to be; Continental, Vittoria, Schwalbe and Michelin in no particular order.
Road Bikes
The vast majority of tyres for these are slick or have very minimal tread (not that it serves any purpose). Most popular sizes are 23mm or 25mm on 700c wheels. The very smallest frames will sometimes use smaller 650c wheels though to keep the geometry right. In general roadies tend to go for the fastest rolling tyre that still resists punctures, and most manufacturers have such a tyre in their range. There are also tyres with high puncture resistance, though they are usually slower. Since tread pattern doesn't affect anything, the choice is simply down to speed, puncture resistance and brand (each use their own compounds).
Note that not all road bikes will currently accept 25mm tyres due to frame clearance issues. Check before you buy.
Hybrids
These commonly use wider and more comfortable tyres than roadies, 28mm or 32mm are most common. Road based Hybrids will use 700c wheels, MTB based bikes will use 26inch wheels with fatter tyres. Such tyres will tend towards higher puncture resistance. Again, tread makes no difference on the road, it's only of any use if you go off-road. Light off-road use is something a Hybrid is well capable of, so it's worth getting appropriate tyres if you are venturing down towpaths or gravelled trails. Cyclocross tyres will usually fit 700c Hybrid wheels, these will have coarser MTB style tread for dealing with tougher off-road conditions.
Note that "Fast Hybrids" (AKA Flat-bar Road Bikes) are often fitted with 25mm road tyres and might not accept 28mm or above.
Mountain Bikes
Designed for full off-road use, they have fat tyres with blocky tread patterns for gripping gravel, wet grass, mud etc. MTB's are now available with three different wheel sizes, 26inch, 29inch and 650b. Then you will have different treads to suit different surfaces (a tyre to deal with mud will have a very coarse tread, one a firm trail will be much less so) but these treads make MTBs slow when on the road. If you do a fair bit of road riding, you can get slicker and lighter road tyres that will solve that problem.
There is a huge amount of variety in MTB tyres, but there are some all-rounder tyres on the market too that will go just fine in all but the most extreme situations.
Hope that helps