Saltwater Suzi and Cap'n Larry's advice for the cash-starved boater
Boat and Ship Rigs
Modern Rigs
Masthead Sloop  - the most popular rig, having a single mast with two sails.  The forward sail is called a jib.  The aft sail is the Mainsail, or Main.
Cutter Rig -  sometimes called a Cutter Rigged Sloop, is rigged for two sails forward of the single mast and one sail aft.  The forward sail is called the Yankee, the inner forward sail is called the Staysail.  The aft sail is the Mainsail, or Main. Usually the mast is located further aft than the sloop to leave more room in the foretriangle for the two sails.
Fractional Rig Sloop - Like the masthead sloop except the forestay is not attached at the top of the mast but generally 3/4 to 7/8 up.  This allows for bending the mast (by adjusting the backstay) to control the shape of the main.
If the descriptions above are unfamiliar to you, here is a basic diagram of sailboat parts.  For a more complete description of the many sailboat parts see our page called,"How to Sound Salty"
How is knowing this stuff going to make you a more Frugal Mariner?  It's probably not going to make you more Frugal, but it will help make you more of a Mariner.  If you don't already know this stuff, you should spend some time learning it.
Ketch - has two masts, the forward mast (main mast) is taller than the aft mast (mizzen). The mizzen mast is forward of the rudder post. Ketches can also be cutter rigged. The aft sail is generally just called the mizzen, not the mizzen sail.  Frequently on ketches, there is a stay connecting the tops or near the tops of the two masts.  This is called the triatic stay.
Yawl - like the ketch only the mizzen mast is located aft of the rudder post.  The standard joke is "a Yawl is just a southern ketch."  Remember you heard it here first.  Both the ketch and the yawl can fly a jiblike sail between the masts called a mizzen staysail.  The advantage of the ketch and yawl is they can carry the same square footage of sail as a sloop with sails which are more easily managed by a smaller crew.
Schooner - has two (or more) masts. The aft mast is the mainmast and is as tall or taller than the forward mast(s).   The varieties of schooner rigs are too numerous to mention.  The typical modern schooner is often rigged as above.  Currently, multi-masted boats, ketches, yawls or schooners are seldom manufactured.  They are included here under modern rigs because there are still many in use today which were built during the last several decades.
Gaff rigged sloop - you won't see many of these today, but occasionally, you'll see maybe a custom built designed or built by someone with a more traditional opinion of how boats should be built.  A gaff rigged boat will give the boat a consideralbly larger sail area for the mast size, though such a boat probably will not go to weather as well as a conventional sloop, it will easily make up for it on a broad reach or a run.
A sailor never shaves his beard.
He seldom bathes as well.
If he has a girl in every port,
They have no sense of smell.
Cap'n Larry sez:
Cat Boat - has a single sail on a mast which is set far forward.  These are usually very easily single-handed, though they may not go to weather as well as a sloop.  Some of the recent designs use unstayed masts.  Many smaller dinghy sized boats are cat boats. 
Traditional Rigs
As I am sure you already know, there are many types of ship rigs, both traditional and modern.  If you feel like researching, here is a list to help get you started:
1 Flying Jib 17 Main Upper Topsail
2 Outer Jib 18 Main Lower Topsail
3 Inner Jib 19 Main Course
4 Fore Topmast Sail 20 Mizzen Royal Staysail
5 Fore Royal 21 Mizzen Topgallant Staysail
6 Fore Upper Topgallant Sail 22 Mizzen Topmast Staysail
7 Fore Lower Topgallant Sail 23 Mizzen Royal
8 Fore Upper Topsail 24 Mizzen Upper Topgallant
9 Fore Lower Topsail 25 Mizzen Lower Topgallant
10 Fore Course 26 Mizzen Upper Topsail
11 Main Royal Staysail 27 Mizzen Lower Topsail
12 Main Topgallant Staysail 28 Crossjack, mizzen course
13 Main Topmast Staysail 29 Spanker
14 Main Royal 30 Foremast
15 Main Upper Topgallant Sail 31 Mainmast
16 Main Lower Topgallant Sail 32 Mizzenmast
Barca-longa Felucca Man-of-war Ship of the Line
Barque Fifie Mast Aft Rig Sixareen
Barquentine Fluyt Mersey Flat Sgoth
Bermuda rig Fore & Aft Rig Multihull Sloop
Bermuda sloop Frigate Nao Sloop-of-war
Bilander Full Rigged Ship Nordland Smack
Brig Fusta Norfolk Punt Snow
Brigantine Gaff Rig Norfolk Wherry Square Rig
Caravel Galeas Pilot Cutter Tall Ship
Carrack Galiot Pink Thames Sailing Barge
Catamaran Galleon Pinnace Trailer sailer
Catboat Gunter Rig Pocket Cruiser Trimaran
Clipper Hermaphrodite Brig Polacca Vinta
Dutch Clipper Herring Buss Pram Wherry
Cog Hoy Proa Windjammer
Corvette Jackass-barque Punt Windsurfer
Cutter Junk Razee Xebec
Dhow Ketch Sailing barge Yacht
Dinghy Longship Sailing hydrofoil Yawl
East Indiaman Lugger Schooner Yoal
The most common of traditional ship rigs

A brigantine is a vessel with two masts, only the forward of which is square rigged.

A brig is a vessel with two square-rigged masts

A barquentine (also spelled barkentine) is a sailing vessel with three or more masts; with a square rigged foremast and fore-and-aft rigged main, mizzen and any other masts.

A clipper was a very fast sailing ship of the 19th century that had multiple masts and a square rig

A frigate is a warship rigged similarly to a clipper.

A schooner is a type of sailing vessel characterized by the use of fore-and-aft sails on two or more masts with the forward mast being shorter or the same height as the rear masts.

A top-sail schooner is a variation on the schooner, with square sails at the top of the foremast.
"Climbing the Ratlines"  Photo by Elliot MacDonald  (AKA Cap'n Larry)
Taken during the 2008 Chesapeake Bay Schooner Race.