Brave Bit of my specialist subject 🙂, although I’ve not done much in the last couple of years either.
‘Recurve’ = a single curve with two reverse curves at either end with no wheels or slats or anything
‘Barebow’ = no complicated sights or stabilisers.
‘Olympic recurve’ (so with sight mechanisms and stabilising weighted arms) is (obviously) what they do in the Olympics and is kind of the standard competitive form. ‘Compound’ is when the bow has cogs and/or cams that effectively multiply the force the bow puts behind the arrow - you generally don’t pull a compound back with your fingers because the forces involved are risky, you use a mechanism like a hook with a trigger instead.
There is also field archery (shoot at targets placed along a pathway generally though woodlands) and clout archery (which is descended from archery in warfare and basically involves landing arrows into a target painted on the ground about 150-200 yards away).
If it’s something you want to look into further, the Archery GB web site has a ‘club finder’ section and clubs generally do beginners courses that go through the basics.