For those not familiar with compression testing or leak down testers here's a nice wee video which covers most of the points. I would only add that, during the leak down test, he turns the engine crankshaft, while the tester is issuing air to the cylinder, until it shows pressure (thus confirming both valves closed) I was taught to set the engine up TDC on compression stroke before applying pressure via the tester. I prefer to do it this way because, as they age, bores wear more at the top of the stroke than further down. So by testing at, or near to, TDC you're going to pick up the "worst case scenario". There is a problem to be aware of though. When you admit the air - commonly at 100psi - unless the crank is very accurately at TDC then the pressure will drive the piston down the cylinder and rotate the crank (with some force, so beware fingers getting trapped in drive belts etc.) I find that if you position the piston just before TDC and hold the crank with a long handled power bar then it's very easy to stop the crank turning backwards with your hand - at or near TDC there's almost no angular leverage exerted by the piston/con rod on the crankshaft so very little force needed to resist turning. At college the instructor insisted we position the crankshaft exactly at TDC so the con rod was vertical and thus the crank wouldn't rotate when pressure was applied - really quite difficult to do. It was a real revelation when I did my first one in the garage workshop and the old fellow who was looking after me showed me how to do it with the breaker bar. Anyway, here's the video:
One thing I've realised he doesn't mention is doing a dry and wet compression test and when you'd use them. A dry test is performed exactly as he shows. If you get maybe one cylinder with a poor result, it's worth removing the tester and, using an oil can, give a couple of good squirts of oil into that cylinder, stick a rag over the plug hole and spin the engine over on the starter a few time to expel excess oil, reinstall the compression tester and repeat the test. If the pressure is better then it's very likely the liquid oil will have partially sealed up the fit between rings and bore this giving temporary increase in pumping ability. No pressure increase usually indicates the problem is more likely to be valves or maybe even a hole in a piston? Always worth checking if any bubbles come up in the radiator coolant expansion tank with the engine running which would point to a leaking/blown head gasket.
For us driveway grease monkeys, the compression tester is a cheap and very usable thing to own. The leak down tester tells you much more but needs a source of compressed air, which many of us won't have. Luckily I have compressed air so I'm not restricted in this way, but still find both testers useful in their own way (I have to borrow the leak down tester though as I use one so little it doesn't warrant buying one. I actually made my own compression tester!