I have lots of saws...
Table saw - Ridgid 10" contractor's saw, bought in 2000, just after I moved into my house. This is the most used saw in the shop. I'd had two before this one. The first was an inexpensive one that I got when I in grad school. It took a lot of hard use before the motor finally died, especially since I let my landlord use it while renovating the house in Slocum. The second was a Craftsman that he bought and left in my care when he and my landlady split up, partially as recompense for killing mine. He eventually took it back not long before I bought my house. The best thing about the Ridgid is that is has one of them snazzy new-fangled rip fences that is terribly easy to set up and get parallel to the blade. That was always a chore with the crappy rip fences that came with the others. Makes me actually eager to use it. The other nice feature is the mobile base that lets me move the thing around easily, a necessity when you don't have a lot of room.
Radial arm saw - Craftsman 10" radial arm saw, bought ten or twelve years ago. The radial arm saw is great for making precision cross cuts, and for doing things like dadoing and rabbeting. It's a more specialized tool than a table saw, but at the time, I had my ex-landlord's table saw, so I got this to complement it. Second coolest tool in the shop (the first being the planer, but that's a different entry).
Miter saw - Craftsman 10", a floor model I picked up in 2001. A miter saw is sort of a poor man's radial arm saw. The nice thing about it is it's portable, so if you're working in the attic, or outside, you can bring it with you. I really haven't done much with mine. Since it was a floor model, it didn't come with a manual, and it turned out it was out of alignment, so the cuts weren't truly 90°. I've since gotten a manual, but I have yet to actually align the thing.
The nice thing about the first three saws is that the same blades fit all three, which means I can do things like set up the table saw for ripping (cutting with a coarse blade along the grain), and the radial arm for cross cutting (cutting with a fine blade across the grain), rather than constantly having to swap blades. Very handy, that.
Band saw - Delta 8" bench-top model, bought eight or ten years ago. A band saw blade is a long loop of flexible metal that runs between two pulleys. It's good for cutting curves, and resawing wood (cutting thinner slabs from thick ones), although the one I have isn't really powerful enough to do the latter. And I have trouble staying near the lines when cutting curves. Someday I'll get a full sized one.
Circular saw - Craftsman 7 1/4", bought a couple years ago. The standard construction worker's power saw. I use mine mostly for cutting plywood panels. I've found it to be far easier to cut plywood by using a circular saw and a long, sturdy straight edge than trying to muscle a 4' x 8' sheet through the table saw.
Jigsaw/saber saw - Another Craftsman (see a trend developing?). These used to be called saber saws, because a jigsaw was actually something else, a bench tool with a very thin blade. Now what used to be called a jigsaw is called a scroll saw, and saber saws are called jigsaws. Got that? I got mine as a Christmas present when I was in grad school. Useful for cutting curves, although as noted above, I suck at cutting curves. Be that as it may, if I could only have one power saw, I would probably pick a jigsaw, just because it can do pretty much any kind of cutting one would need to do. It may not be the best choice, but it will probably get the job done adequately. (My mother built our old kitchen table using only a jigsaw and a screwdriver.)
Chain saw - Craftsman again. (No picture, because everyone knows what a chain saw looks like.) Used for very fast, very rough cutting. Chain saws are not what you'd call finesse tools. Mine is electric rather than gas, which means I'd need long extension cords if I wanted to terrorize my neighborhood.
Hand saws - I have a lot of these, all different shapes, sizes, and purposes, that I've collected over the last thirty years. Some I bought and some are hand-me-downs from the ex-FIL. There are a couple of hacksaws for cutting metal, a keyhole saw for tight places, and a wallboard saw for, well, cutting holes in wallboard. A coping saw is the hand-powered version of a scroll saw, and a jeweler's saw, which is what happens when a hack saw and a coping saw love each other very much. Dovetail saws are used for very precise finish work, as are their smaller brethren, razor saws. For outdoor work, I have two bow saws, and a folding pruning saw (aka, camp saw). To be honest, I haven't used either bow saw since I got the chain saw. The pruning saw is perfectly adequate for anything I would still use the bow saws on around my yard. Finally, there is my trusty, thirty year-old standard hand saw, which I probably haven't used in twenty-five years. First of all, there's the whole not being able to follow the line thing. I never mastered the technique of making straight cuts with a hand saw. The blade flexes and wobbles, and suddenly the cut has wandered a quarter inch to one side or the other from the line. Plus, sawing wood by hand is hard work. Whenever I have a choice between using a hand saw or a power saw, I always go with the power saw.