I think I've heard this before referred to as an interference design versus a non-interference design. An interference engine will sustain major damage in the event of a broken timing belt, whereas a non-interference engine will simply stop working until the timing belt is replaced.
Interference engines, when the belt or chain breaks, will run into major issues from the valves and pistons trying to occupy the same space at the same time. Hence the term "interference," since the valves and pistons will interfere with each other. If this happens, you're looking at a complete engine rebuild.
Noninterference engines, since there's enough room for the valves to be open and the pistons to be raised at the same time, will simply quit running. You're looking at a little more labor than a straight belt/chain replacement, but not by leaps and bounds.
Timing belts usually have replacement intervals around 80,000-100,000 miles. You want to stay on top of that, since a belt won't give any warning before it lets go. Honda is pretty notorious for using belts in interference designs- if everything holds, their engines are great, but you have to keep up on the maintenance (they usually recommend swapping the water pump at the same time).
Timing chains generally have no replacement interval- you don't worry about it going unless it begins to chatter on you. Since they are lubricated with engine oil, they last a good long time. 200,000+ miles is not unusual at all. Many automakers are going to chains for that reason; they can be noisier than belts, but I've never noticed that to be an issue.
These are separate from drive belts/serpentine belts. Those drive the engine accessories, like the power steering pump, water pump, alternator, and A/C compressor. Sometimes there is a couple belts ("drive belts"), but nowadays there's usually just one ("serpentine belt"). Generally, replacement intervals on those are 30,000 miles on the low end, and maybe over 50,000 miles. Belt quality is far better than it ever has been and they last a LONG time. Drive belts can just be watched for wear; if they squeal, it's time to change them (which can take 5 minutes in a driveway by anybody or a couple hours in a shop by a trained mechanic, depending on design; I've seen it both ways). Serpentine belts are usually easier to change out than drive belts, since there's generally an automatic tensioner that's easy to get to, and you can often thread them in from above the engine.
Hope this helps clarify things.