Handwraps of Awesome

Awesome gloves that are made entirely out of wrapping bandages or cloth around the hands and knuckles. Often seen during poses flying free or being wrapped using the character's mouth. Boxers tend to wear these since normal boxing gloves aren't cool enough for fiction. Application usually takes place in the montage before a fight.

When you see these on a female character, don't be surprised if she wears a Sarashi as well.

Truth in Television, as wearing hand wraps is generally a good idea. The compression helps reduce metacarpal fractures and braces the wrist, which is very important when your blow doesn't land dead on. That and boxing gloves are really uncomfortable without that extra layer. But not all the time...

Examples of Handwraps of Awesome include:

Anime and Manga

Sasagawa Ryohei of Katekyo Hitman Reborn is never seen without handwraps.
Rock Lee in Naruto wears those. He actually unwraps them during a fight, to restrain the enemy for his Meteor Move.
- Sasuke also has these for a short time in the Chunin Exam arc. A flashback shows that Rock Lee's mentor Maito Gai had them in his youth.
Yuu from Holyland takes to using these when he's not using his homemade pair of plastic hand guards.
All Muay Thai fighters wear these in Kenichi: The Mightiest Disciple. Since Kenichi's custom-made training uniform (introduced in Chapter 145) was patched up from pieces of outfits for various martial arts (which reflects various fighting styles of his mentors), it also includes a pair of such bandages.
- Ikki Takeda also wears these on a constant basis since, well, he IS a boxer.
Ichigo dons these occasionally in Bleach.
Black*Star from Soul Eater seems to have swapped his old gloves for a pair of these after his last rematch with Mifune.

Comic Books

Wildcat from The DCU is typically seen wearing these, which makes sense given that he's a boxer.
Kendra Saunders/Hawkgirl in Justice Society of America, except in her case they go halfway up her arms to hide the cuts on her wrists.
The Batman villain Killer Croc sometimes wears these. Examples are Batman Hush and Batman: Arkham Asylum. (In the concept art, at least. His in-game model wears a torn prison uniform and broken handcuffs.)
Katchoo in Strangers in Paradise wraps her hands when she knows she'll have to fight. So do her big sisters, the mafia bodyguards.
Daniel Rand, The Immortal Iron Fist, has wrist/forearm wraps in his white costume, with taped fingers as needed.

Film - Live Action

Marcus Wright from Terminator Salvation.
In Hot Shots: Part Deux, Topper Harley wears these in his Dim Sum match early in the movie.

Professional Wrestling

Handwraps are very common in pro wrestling, especially with wrestlers who have a boxing or martial arts background.
CM Punk's handwraps, in particular, are a very visible part of his usual attire. He always marks the wraps with an X on the back of his hands, a symbol of his commitment to the Straight Edge lifestyle.
- While Punk marks the "X" for straight edge, it's also worth noting that a boxing referee (or a referee in any type of legitimate martial competition for that matter) will mark an "X" (or some other emblem, an X is the most popular) on the wrappings around a boxer's hands after he has inspected said wrappings to make sure they are technically legal and not hiding anything that would aid him in the fight (thin sheet of aluminum, razor blade, what have you). In effect, this serves as a Shout-Out to both elements of his character.

Video Games

The Scout from Team Fortress 2.
Adon, Akuma and Sagat from Street Fighter, Sagat being a special case as he wraps his hands and feet to go with his fighting style Muay Thai.
In the recent Tekken games, there's Steve Fox, a British boxer. His 'main' costume has classic boxing gloves (along with very loud shorts), while his second costume is 'civilian' clothes, with fistwraps. IIRC, Heihachi Mishima also often uses fistwraps, while his descendants seem to prefer half-gloves. Guess he's just 'old school'.
Kano from Mortal Kombat wears these.
In City of Heroes fist wraps are one of the costume options for gloves.
In Lost Saga, if the Boxing Champ hero loses his gloves, his hands are shown to be wrapped like this.

Web Original

Rottweiler, a street-level martial arts hero from the Global Guardians PBEM Universe (think Wolverine without the claws but with a much sunnier disposition), wears handwraps as a matter of course. Cestus, a member of the supervillain team Hard Corps, does the same.

Real Life

Tefillin, a part of Jewish ritual prayer garb. Part of Tefillin is a piece of black leather that ends being wrapped around certain fingers.