There are even more shadings than that; Glock refers to their proprietary trigger as something else....
Essentially, "single-action" autos (best example-the M1911) only fire from the full-cock position. Normally a manual safety can be applied, and there may (as in the M1911) be secondary safeties as well.
You can carry these weapons in several "modes", such as "cocked and locked" (ready to go except for the manual safety), "condition two" (hammer down on a live round-requires manual cocking of the hammer), etc.
Double action weapons, such as the Walther P38, PPK, etc, allow firing the first shot by simply pulling the trigger through a long, hard first cycle, similar to a double action revolver. On subsequent shots, the trigger remains back, and has a light pull similar to a normal semi-auto. Normally, these pistols have a "drop the hammer" safety, which lowers the hammer onto a live round.
Advantage-you can carry the weapon ready to go, and just pull the trigger.
Disadvantage-some find the change of trigger position disconcerting, and the safety mechanism complex.
Double-action-only pistols simply eliminate the change described above. Each pull of the trigger cycles the hammer/striker, much like a double-action revolver.
Advantages-the trigger pull is always the same, and it's very safe to carry.
No safety devices to fool with.
Disadvantage- precision shooting is somewhat difficult, depending on design.
Glock has an action which functions like the full-time-double action, but works a bit differently. No external safeties whatever. Pick it up and shoot. There are a number of internal safeties. Advantage-simplicity.
Disadvantage-some find Glock's trigger action a bit annoying and difficult to shoot with precision. (they do make "target" models)
You will find ferocious advocates of all these systems, and all are in use in various military/police forces all over the world.
Personally, I think the Glock is perhaps the easiest weapon for those with revolver experience wanting to "transition" as we say.
All of these weapons can provide fine service, and one should train with whatever you decide on.
Always a good idea to go somewhere that will let you shoot a variety of weapons to see what suits.