A watch clasp is a mechanism that attaches the two ends of a watch band together when it's on your wrist. They're often used on metal bracelets, but you can also get them for leather straps as an alternative to a buckle.
There are many different types of clasps, but there are a few general types that are very popular; deployant clasps, deployant clasps with an extra catch and butterfly clasps.
The phrases clasp/deployant clasp are often used to refer to the folding mechanisms used on metal bracelets, but mechanisms used to attach watch bands/straps together more generally may also be referred to as clasps (so include buckles). This also varies depending on which country you're in; in the UK you may come across deployment rather than deployant for example.
Clasps tend to be fastened to both sides of the watch band and when in their detached position they still attach both sides together but more loosely, so it wraps around your wrist still but would fit around your hand to remove it. The clasp may be fixed to the band/strap or detachable.
Generally, when the clasp is closed you should have enough space to fit a finger between your wrist and the clasp. Some clasps require this to be able to remove them. When storing/moving watches it may be best to keep clasps closed to avoid damage.
Buckles vs Clasps
Leather watch straps often come with buckles rather than clasps, where one side of the strap has holes down the middle and the other has a rectangular hole at the end with a pin. The side with the holes feeds into the other side when you put it on your wrist and it's attached using the pin. Most metal bracelets will use a clasp of some sort instead of a buckle, and leather straps may also use them.
On metal bracelets the clasp is often fixed to the bracelet, so you may need to go to a watch professional to get it adjusted. When attaching clasps to leather straps, the clasp may be fastened to the leather on both sides or maybe just one with the other side attached using the buckle holes to allow it to be adjusted easily. In any case, make sure it's fastened securely. If attached using buckle holes the clasp pin for example may be shorter/looser than the pin on a buckle.
Buckles are generally comfortable and quite tough, but you can't really use them on a metal band/bracelet. Also, for buckles you often need to bend the strap to put the watch on/off and this can cause the strap to wear down/crack over time, or you might find the buckle digs into your wrist and causes discomfort. Clasps are generally heavier than buckles and larger, but they work differently and come in different shapes, so you may find one or the other more comfortable.
A clasp would normally be made of metal and rest against your wrist like a buckle, but the mechanism they use means they're much longer and have varying widths. If the clasp aligns with your wrist shape well it might be quite comfortable, or you might find that it feels heavy, doesn't fit well or causes discomfort. They can also catch on arm hairs and/or your skin. Or you might just prefer the look of either buckles or clasps. So, in general both buckles and clasps have pros and cons.
Deployant Clasp
A deployant clasp has three pieces of metal that fold on top of each other, where the bottom two pieces are attached together on a hinge so that when done up the attached bracelet fits on your wrist but when detached it opens in one direction and leaves enough space to remove it. They're often made of stainless steel and different types of them are widely used on watches with metal bracelets.
When the clasp is done up it will typically clip into place and need to be unclipped to remove it. This might be done using a button on the sides of the clasp, where you'd press both at the same time to detach it, or you may need to detach it by putting a finger between the bracelet and clasp and apply pressure. Be careful not to damage the clasp when removing it. If removed with your finger be particular careful and it's important to leave enough space under the bracelet to fit your finger so you can remove it and to prevent it coming off by accident if you move/bend your wrist.
Deployant Clasp With Catch
Deployant clasps with an extra catch on the outside of the watch bracelet are also quite common, where the catch needs to be unclipped before the clasp can be detached so it's less likely the clasp will come off unintentionally. You can also get this with a third mechanism, where a button would need to be pushed before the catch can be undone.
Butterfly Deployant Clasp
A butterfly clasp is similar to having two normal deployant clasps, where it has two hinges rather than one and opens in both directions rather than just one. These can be quite thin and mostly below the watch bracelet so give the bracelet a nice aesthetic.
Like a normal deployant clasp you might detach this using your fingers or by using buttons on the clasp. If detached using fingers these can be quite tricky to open/close.