An open diff is whats most common in vehicles. That setup is the ring gear with a carrier and spider gears. Only 1 wheel is powered with this type of diff and its the one with the least amount of resistance. This means that if you have one wheel on gravel and one on ice, the one on ice will get all the power and the other wheel won't spin at all. It also gives the smoothes performance while turning, but not good for snow/ice/mud.
A LS (limited slip) is a carrier, typically with no spider gears, that contains a series of clutches. These clutchs engage when one wheel starts spinning more than the other. These clutchs then transfer power to the other wheel. They don't have quite as good a road manners as an open, but most times aren't bad. They are also known to fail at bad times like when you're stuck in mud, there is too much resistance to overcome and instead of getting the power to the other wheel the clutchs just spin. Its good for mostly onroad driving and some light offroad.
A locker is any kind of a positive locking device. There are many different kinds from a drop in (replaces the spider gears) to a full spool, or a spool. The drop in (commonly called a lunch box locker) are the cheapest but have unpredicatable nature on the street when turning. They are also the most prone to failure. The full spool are the most expensive but have better onroad manners and are more reliable usually than the lunch box style. The spools either replace the spider gears (mini spool) or the entire carrier (full spool). These are solid pieces of metal that lock the 2 axle shafts together and they can't turn independantly of each other at all. They have the worst street manners, but are great for offroad. The spools are relatively cheap though, my mini spool for my 9" cost me $16. I do have a Lincoln locker in my Ranger axles currently, which is where you weld the spider gears together and then weld them to the carrier. Its definetly NOT a real good way to do it, but my Ranger never sees the street. Besides that the axles that are in it aren't in the greatest condition anyhow and I have a D44/9" combo waiting to go in.
Unless you're doing alot of offroading I would stick with a LS or open diff.