The timing belt is driven by the engine crankshaft, and it turns the engine’s camshaft(s) (you may have more than one). The camshafts then push the engine’s valves open at the appropriate time, which lets fuel into the engine cylinders and lets exhaust out at the end of the engine’s cycle. It’s called a timing belt because it controls the timing of these various valves, when they open and close. They used to use a chain, but these days they use belts so that they can make the engine quieter and a bit lighter. The belt has “teeth” on it so that it keeps the spinning of the engine crankshaft (which is driven by the pistons) and the camshafts in perfect sync.
There are two types of engines, interfering engines and non-interfering engines. An interfering engine means that when the valve moves into its open position, it is in the same place inside the cylinder where part of the piston is. In a non-interfering engine, when the valves open, they open into a space where the piston never goes. I’ll take the previous poster’s word for it that you have an interfering engine. What this means is that if the timing belt breaks, then the valves may end up open when the piston comes up to the top of the cylinder. This usually results in the valve being obliterated, accompanied by severe damage to the piston and the inside of the cylinder. In a non-interfering engine, if the timing belt breaks, then the engine just stops running. Since the valves never move into the same area where the piston travels, all that happens is that the engine stops.
You have an interfering engine, which means that if the timing belt breaks, there’s a whole bunch of stuff inside the engine that can get very badly damaged. Basically, you can just about ruin your entire engine.
Changing the timing belt is a big job, because you basically have to disassemble the entire front of the engine to get to it.
When you look at the engine, you see the serpentine belt, which drives the alternator, power steering pump, and the compressor for your car’s air conditioning (that’s why the call it a serpentine belt, it snakes its way around all of those devices, plus maybe the water pump pulley and one or more tensioners). To change the timing belt, everything I just named has to come off of the engine, along with the pulleys and things that they are attached to. Then you have to take off the front cover of the engine. Now you can finally get to the timing belt. Usually, the mechanic will change the water pump while he’s in there, since you’ve already paid for the labor of disassembling the entire front of the engine and the cost of the pump is relatively cheap. You wouldn’t want to cheap out on the water pump and not replace it, and then have to spend all of that labor again later to replace it when it does fail. Once the timing chain and water pump are replaced, then the entire front of the engine has to be put back together.
Needless to say, this job costs a bit more than swapping out a new serpentine belt, so be prepared for a somewhat hefty repair bill. Again, though, not doing this repair can destroy the entire engine, and fixing that may be more than what the car is worth.
Since timing belts are made out of rubber, and even under little old lady minimal use that rubber will eventually rot and fail, it’s time to replace the timing belt.
ETA: Here’s a youtube video that explains changing out the timing belt on Toyotas similar to yours: