Hi Matt;
Yes, a gap bed lathe has a piece of the bed (ways) that can be removed. It sits right under the chuck. You can take it out so the jaws on the chuck can be opened up all the way, or to put a larger chuck or face plate on the spindle. It also lets you do larger diameter flat pieces on what is basically a regular sized lathe, (whatever "regular" size is). Like, normally a lathe may be able to turn only 9" diameter, but if you have a gap, you could turn a 14" flywheel for a car. Things like that.
An engine lathe is what you will usually find in a job shop, where all manner of work will come in. It does all the regular chores at a good level of accuracy.
A tool room lathe is similar, but generally made to tighter tolerances and often have specialized chucking options for making somewhat more specialized items. If you are making tools that will be used to make other tools, like the Starrett company does, you use a tool room lathe. They can be set up to do very accurate work in batches. They are often setup to do very fine threading that an engine lathe operator would have a hard time duplicating. Usually the machine will be held to a tighter tolerance in the areas of the spindle, ways, leadscrews, and like that. They often have a rapid threading option, and the ones I've seen are kind of scary to watch. They will zip through a thread really fast, right up to a shoulder, then stop immediately and return to the start position.
An engine lathe can do most of the things a tool room lathe can, but the engine lathe would be considered the jack of all trades, where the tool room lathe would be the specialist.
There are a number of differences I'm leaving out, but you should get the idea.
Dean