There are quite a few styles of straps anyone could use on their watch. But there are also different types of buckles and clasps that a person could use on their strap as well. I will go through some of the more common straps and buckles as well as any extra functionality.
Quick-release strap
Although this can be seen on any type of strap it can be a feature that a lot of people want on their strap. As this mechanism allows a person to quickly change their strap without using a spring bar tool. This is very easily identifiable by a little knob that appears right at the end of the strap where it connects to the lugs. This is a good feature for anyone who wants to change their watch straps daily or whenever they decide to wear their watch.

Deployant Buckle
A deployant buckle is most commonly seen on leather straps on higher-end watches. However, you can usually buy the buckle for about $15-$20 and easily attach it to the end of your current leather strap. People usually add this buckle to help avoid creasing on their leather strap as the person will no longer have to push in the leather into the buckle and loop. I would recommend anyone to get this buckle on their leather strap to prevent it from creasing and also gives a much nicer look on their strap.

Leather Straps
Leather straps are very common but the variation and type of leather can be vastly different. Leather straps can come quite cheap but also can be quite expensive depending on how expensive or exotic the material used for the leather is. Sometimes purely because the brand name is stamped onto the leather can also affect the price even if the quality does not differ.
Leather straps can come plain, stitched or a combination of both while being made out of an exotic leather material such as crocodile or ostrich.
Plain leather strap Stitched leather strap Crocodile leather strap
Perlon Strap
These straps are quite easily identifiable by the woven pattern look. It is usually made of nylon material, however personally I do not think this is a very popular choice in straps. I will admit it does have a very distinct look and goes very well with a wide array of watches.

Aviator Strap
Much like the name suggests these straps are usually seen and work best on pilot watches. These straps are very easily identified by the very distinct rivets usually seen right next to the lugs of the watch. These rivets used to be for functional purposed but now they are more placed just for the aesthetic.

Rubber Strap
Rubber or silicon straps are mainly characterised as the sporty variation of straps. One extreme example of a rubber strap being used on very expensive watches is with the manufacturer Richard Mille. These watches can go easily from high six figures to a million dollars. This is a bit of an extreme example but the point is that the the sportier the watch the higher the chances the manufacturer or brand will pair it with a rubber strap.

Mesh/Milanese Strap
They both look quite identical, to describe it in words think of it as a more chain mill pattern. If you are looking for a mesh strap that has very a dense construction try searching for a Milanese strap. These straps are known to be quite comfortable to wear and also gives the watch a very professional look.

NATO Strap
NATO straps are usually made out of nylon and the main discerning feature of it will be the iconic two strips of colour seen on a NATO strap as well as the multiple loops. These straps can drastically change the whole look of the watch and also be a good way to dress down a watch to make it appear more casual or sporty. This strap also gives additional security since it is made up of one single piece. Therefore, if it breaks from one lug it will still hold on to the watch.

Zulu Strap
They are very similar to NATO straps but are usually made of thicker materials and can also be with a different number of loops than a NATO strap. Due to the thickness, it will need to work with the current spring bars of the watch or it might risk breaking.

Jubilee Bracelet
This design was introduced by Rolex, and has become an iconic pairing with many Rolexes. It is very easily identified with three highly polished centre links in between the much bigger unpolished side links.

Oyster Bracelet
Much like the jubilee bracelet, the oyster bracelet was introduced by Rolex. This bracelets style has been copied all over and its distinct look is more functional. As compared to the other bracelets it is a lot stronger due to its simple and effective design. The bracelet has three distinct metal links all similar or the same size with an additional wider centre bar to make the strap more durable.

President Bracelet
This bracelet again introduced by Rolex has a very similar look to the jubilee bracelet. However, with this bracelet the links are shorter making it look a lot more compact. It still has the exact same look of the matte finished side and the polished centre link seen on a Jubilee bracelet.

Engineer Bracelet
Finally, a bracelet not made by Rolex, the engineer bracelet is the last bracelet I will be talking about in this post. This bracelet is usually seen in some diver watches or chronographs. They are quite chunky and are usually spaced with 5 links going across the strap.
