Yes. Sleeves have names. There is a history behind the styles of sleeves and how they are cut and sewn. In this post, I give the sleeves loose definitions and show pictures of the different named sleeves. This will give you a pretty good idea of the variety of sleeves available so that you can add variety to your wardrobe.
A bell sleeve flairs at the bottom like a bell, very much like bell bottom pants. This sleeve is my current favorite. I love the flow of the sleeves and how feminine they look.
A kimono sleeve is a sleeve cut in one piece with the bodice. Examples are shown below. There are many interpretations of what this sleeve looks like, and this sleeve does not have to look like a traditional kimono.
A raglan sleeve is a sleeve that extends in one piece fully to the collar, leaving a diagonal seam from underarm to collarbone. It is named after Lord Raglan, the 1st Baron Raglan, who is said to have worn a coat with this style of sleeve after the loss of his arm in the Battle of Waterloo. (Wikipedia)
These are popular with tees, as you may imagine, as well as with a few other styles of clothing. This is common in sports like baseball, but can also be seen in many other styles of clothing.
Off the shoulder sleeve is easy to define. It begins off the shoulder as pictured below.
The cape sleeve. "Cape sleeves are an overlay and usually a sheer extension to the bodice of your dress. They look like an extension to your sleeves. Some of them are inbuilt while some of them are an attachment." Style Craze
This type of sleeve is very dramatic. It's great for special events.
The bishop sleeve.
A bishop sleeve is a long sleeve that is gathered at the bottom with a button cuff, slit and facing. This lesson will teach you the principles of 'slashing and spreading' to add fullness to sleeves, as well as to other areas of a garment. (Fashion University)
Otherwise, it can be described as a full sleeve that is gathered at the wrist.
The petal sleeve is self explanatory. It looks like a flower petal with the overlay on top.
The butterfly sleeve.
Butterfly sleeve is a beautiful sleeve with a puff on top and slightly loose at the hem. It is basically a puffed sleeve but one that doesn't taper off like the normal puffed sleeve, nor ends in a cuff. It has a very elegant look and is usually paired with low cut gowns. (Sew Guide)
The butterfly sleeve looks like the wings of a butterfly. Very feminine. Somewhat flowy, depending on the fabric used. I love the femininity of this style sleeve.
Capped sleeves. These are like extensions of the collar bone. They just "cap" the shoulder. It takes toned or slender arms to pull off capped sleeves.
A puff sleeve is a short sleeve gathered at the top with a cuff and full in the middle. This sleeve seems kind of formal and works well with formal gowns. Although it works well with blouses. I think of this as the princess sleeve.
A cuff sleeve has a fold or band at the end of a sleeve. This is very traditional and conservative.
Bracelet sleeves. To me, these are like 3/4 sleeves. They are just short enough for you to show off your bracelet, watch or other arm accessory.
Open sleeves, cold shoulder or slit sleeve is very self explanatory. Take a look at the pictures if you question. It's basically a hole in the sleeve that is purposely part of the design. It can be small or large. Lately, this has been a big part of Summer fashion.
Batwing sleeves resemble the wing of a bat. They are long under the arm and go tight to the arm, whether at the wrist or half arm. If you have not tried the batwing sleeve, it is super comfy.
Not all sleeves are comfortable or flattering on every body. Give some of these a try to find the variety that fit your body type best. You may be pleasantly surprised.