There are many different kinds of sailboats. Below are a few simple descriptions and some pictures of a few different kinds of sailboats that you may encounter on the water.
This catboat rig is a traditional gaff-headed configuration. It has a very large sail on a very short, but sturdy mast. The mast is very far forward to allow for all of this sail area. There is no Jib, so handling is easier for one person. This is rarely used anymore, but was used a lot in the early 19th century.
The Sloop has a mast that is slightly in front of the center of the boat. It has a main sail and a jib, improving performance over the Cat Rig. The mast is much taller than the catboat rig. The head sail can have more than just a jib sail, it can also have a Genoa, a jib or a spinnaker. These boats are fast and responsive and many racing boats use this configuration.
A cutter has a mast that is further aft than the sloop. This allows for several head sails to be set at once. The Jib would be set on the headstay that runs from the top of the mast to the bowsprit. The other head sail is called a forestay sail and is run on the shorter headstay that runs to a different point in the front of the boat.
A Yawl rigged sailboat is a two masted sailboat. It is much like a sloop since it has a large mast and has a sail attached to the headstay. The difference is that it has another mast in the aft (rear) of the boat. It is much shorter and is called a Mizzen mast. The Mizzen mast is located aft of the rudder. The Mizzen was originally used by single handed sailors because it could be trimmed easily without rudder input. It was also used by commercial fishing vessels from long ago.
The Ketch is very similar to the Yawl because it has two mast. A Main mast and the Mizzen mast. There are differences, come being, the Mizzen sail is larger than on a Yawl, the Mizzen tends to be further back on the Yawl also. The Ketch is popular with long distance cruisers because of how well it handles in many different types of weather and it can also be sailed with just a jib and a mizzen hoisted.
The schooner is a sailboat with two or more masts. Usually there is a large main mast, with a smaller mast forward of that. There could possibly be a Mizzen mast in the aft (rear) of the boat. They were used in the 18th century and were very fast windward sailing boats. They were also used by Pirates and other sailors needing speed.
So there are some of the basic kinds of sailboats. There have been many different types of sailboats, but most are just a slightly different version of one of these. Let me know what you think and if I should add some other kinds of sailboats.