There are some things that can be canned with just a hot water bath (i.e., you put the food in jars, add the lids and rings, cover with water and boil for however long the directions say) - some fruits and vegetable fall into this category. Then there are things that require a pressure canner, which is a specialized piece of equipment that still scares me a bit - and I make and can stocks roughly every three months! I have an old one that has to be watched constantly and the flame twiddled so it stays in the correct pressure range, but I hear there are new-fangled electronic ones that can regulate the pressure automatically.
The biggies: make sure your jars, et al are clean before you start - I run mine through the dishwasher on the sterilize setting, or if I'm using just a few, I boil them. If a recipe says you can hot-water can something but you're not sure, eg, with some varieties of tomatoes, you can use a pressure cooker recipe. You can't go the other way, though: if the directions say it has to be canned under pressure you can't use the hot water method
Once the jars are out of the water bath/pressure cooker you'll hear popping sounds as they cool. That's good: it means the processing forced some air out of the jars and the lids are now sealed (PV=nRT, my favorite equation!). If after they're cool you can press down on the lid with a finger and it doesn't give then the jar is good to store until needed. If there's give, refrigerate it and use ASAP.
There are a lot of web sites that talk about canning, but I'm leery of any that aren't connected with Ball or another canning company or the USDA.