Dress shirts have cuffs of various styles, and though few contemporary cuffs are as noticeable as the ones in the illustration, the style of cuff a man wears says some small thing about him. That is because cuffs are usually chosen to match the formality of an occasion, and deliberately or accidentally adding to or subtracting from the generally accepted formality is a statement of its own.
Shirt cuffs come in three varieties, button, link and hybrid, and each fastens differently. Hybrids, called convertible cuffs, may be closed with buttons or with cufflinks. Being neither fish no fowl, they should be avoided in my opinion and we will say no more about them here.
The least formal contemporary cuff is the button or barrel type that closes with, no surprise, a button or buttons. Ordinary shirts tend to have but a single one of them per cuff, which leaves the wearer with an open cuff if he loses a button, and a slightly misaligned one the rest of the time as the cuff will pivot on one button. Better is the double button cuff in the photograph (one shirtmaker's house style is a three button cuff however this seems to me to be overkill as there is already more than enough buttonning to do each time a man changes his clothes).
Some makers also offer a turnback version of the button cuff, called the cocktail or Bond cuff, that attempts to emulate the look of the linked cuff and, in my opinion, fails to do so.
The more formal link cuff, on the other hand, has buttonholes on both sides that are closed with cufflinks rather than buttons as in the photo above. This is an older form of closing and of course there are a couple of types.
Single cuffs, the original linked cuff, are the standard for white tie and can also be worn with black tie. Plain front white linked cuff dress shirts are also worn with suits by a few aficionados, but never with odd jackets.
The somewhat less formal French, or double, cuff is folded back on itself. French cuffs are are normally worn with suits or black tie. They are usually considered inappropriate with odd jackets.
Now the designs of the links that hold these cuffs vary widely but the principal types are made from either silk or gold. Gold is of course the jewelry version and, as the late Harvey Amies wrote, "If you despise (gold) you had better have just buttons on your shirt cuff." The two gold ends should be joined with a bar or a chain. Most men who think about these things eschew links with only a single finished end because the other side is frankly ugly. Ugly or beautiful, gold links cost from the hundreds to the thousands of dollars.
At the opposite end of the cufflink pricing spectrum is the silk knot, favored by American traditional dressers and others who appreciate their nominal cost. Charvet, the Parisian haberdasher, introduced them about a century ago.
And that is how shirt cuffs are closed.