Over boring the cylinder is a process to ensure the wall of the cylinder is pristine, providing a sealing surface for the rings to mate. Often when an engine gets old and tired, there will become scuffs in the the cylinder wall surface. This is an example of a scuffed cylinder wall:
After boring and honing, the cylinder should look something like this:
In the picture you can see the cross hatching. This is caused from the honing operation and helps the rings "seat" during the break in process. When the term "seat" is used, it means the ring is worn down slightly, allowing it to mate to the cylinder wall, and form a seal which holds compression in the cylinder/combustion chamber, as well as keeping oil out. A secondary purpose of boring the cylinder is to ensure it is completely round. To do this correctly, the shop should use a torque plate, which is attached to the deck during boring and final honing. This simulates the head being attached to the block during the process, so when the head is finally attached, all the little variances are accounted for and the cylinder is as close to round as can be made.
I am assuming the compression ratios will change so what do mechanics do to compensate for the change in cylinder dimensions?
The static compression ratio (CR) can be changed somewhat when the cylinder is bored. This is due to the small amount of difference from the over bore. When doing the calculations for the CR, the new volume from the enlarged cylinder must be taken into account. Even with the small amount overall which is seen when the piston is at top dead center (TDC), there will be a change, however slight. More than likely there will be other factors which will weigh heavier on the CR than just this, such as the dish or dome volume of the piston, combustion chamber size, and thickness of the head gasket.
Do they use the old piston or get a new one?
If you over bore the cylinder, you must use a new piston. When clearances of the piston to wall are measured in the thousandths of an inch (or sub-millimeter) and you bore the cylinder +.010" over (or +.030" or whatever), there is too much gap for the old piston to take up the slack. More than likely, if the cylinder is worn out, it's usually due to the combination of the piston and rings which caused it. The cylinder gives back as well as it takes, so these parts are worn out as well in most cases. New pistons are a given.
What are the disadvantages of over boring a cylinder?
There are only a couple of minor disadvantages to over boring a cylinder. When you bore the cylinder, you have made the cylinder wall thinner. This can lead to over heating in some cases, depending on the block. Some blocks are prone to cracking in the cylinders after an over bore. In such cases you need to ensure the walls are sonic checked to ensure the wall thickness can support your power goals. It doesn't make sense to bore the block, just to have it fail after a short period of time.
Other than that, the only other downside I can think of is the cost of doing the machine work. Realistically, though, it is a lot cheaper in most cases to have the machine work done than it is to get a new block. If you were to get a new block, you'd still have to do the machine work on it. Plus, used blocks are actually better in most cases (as long as they are found structurally sound) due to being seasoned. When a block has gone through heating/cooling cycles several thousand times during normal use, the metal in the block becomes more aligned, which reduces stress risers. I'll just say, the metal becomes happier.
All-in-all, there really is no downside to an overbore, as long as the block can take it. It's usually far cheaper to have this done and rebuild the engine than it is to buy a new one, or in most cases find/purchase a used engine. Obviously it will take longer, but the end product is worth it most of the time.