No need for you to feel that vague unease any longer! Here are the three most common sailing rigs you are likely to encounter on the water.
SloopThe simplest rig, a single mast. It is held in place with a forestay (which keeps the mast from falling over backwards), a backstay (or sometimes two - keeps the mast from falling over forward), and an assortment of shrouds attached to the sides of the boat to keep the mast from falling over sideways.
The sloop carries two sails as "working sails" - that is, the primary sails (I am not going to talk about spinnakers, reachers, screechers, asyms, etc): the mainsail, and the jib. The mainsail is attached at it's leading edge (the "luff") to the mast, and at its foot, to the boom. The jib's luff is attached to the forestay, and it is controlled only with a line to its aft corner (the "clew").
Some would argue that this is a modification, rather than a rig in its own right. A cutter has an additional forestay, mounted lower on the mast and behind the forestay on the deck. A sail has its luff attached to this inner forestay and either has a small boom (club-footed) or not. This sail is called a staysail - it helps to accelerate the airflow between the jib and the mainsail. It is also a useful sail in high wind, when the jib is too big to use.
KetchA ketch is a two-masted boat. The important distinguishing characteristic is that the shorter mast (the mizzen mast) is the aftmost one.
Because there is no way to attach a forestay to the mizzen (it would interfere with the mainsail), the mizzen mast either goes without (in which case the forward shrouds are brought as far forward as possible), or a stay (the triatic stay) connects the top of the mizzen mast to the top of the mainmast, thereby relaying its load to the forestay. There is no backstay for the mizzen, therefore the aft shrouds are swept back as far as possible. On the other hand, since there is no jib on the mizzenmast, there is little forward load on the mast (leaving mizzen staysails out of the discussion).
The combination of the mizzen and the staysail is particularly well-suited to heavy weather, since it keeps the boat balanced with reduced sail area.
Eolian is a ketch, with a triatic stay. Well, technically, she is a cutter-ketch, since she also has an inner forestay and a staysail. Yes, that means we can fly 4 sails. That first picture is Pau Hana (also a Downeast 45, since destroyed in one of the recent Florida hurricanes) - sadly, I don't have any pictures of Eolian with all her sails up.
Like a ketch, a schooner is a two- (or more) masted boat. But with a schooner, the two masts are of equal size, or the forward mast is shorter.
Here is another Downeast 45, Reward which is schooner-rigged. The angle from which this picture was taken shows how well the rig delivers the cumulative slot effect (what that means will have to be the subject of another post, about how sailboats actually work).