Myself, My Avatar

Maybe it's to explore a new world. Maybe it's to be better and stronger than before. Or maybe it's a mistake and it leaves the person Trapped in Another World.

Regardless of why, this trope involves a person who can be (or who is) downloaded into a remote interface to interact with others or another world better (or at all). This other self is otherwise an empty shell that does nothing. Sometimes when the remote version of himself is damaged or killed, he is hurt/killed back at home or in 'reality', but the idea is, while the main body 'sleeps', a different body acts and works remotely by his commands. This is used to play games, interact with computers, or interact with an alien world.

Unrelated to "(Do You Wanna Date My) Avatar".

Examples of Myself, My Avatar include:

Anime and Manga

This happens to Tsukasa in .hack//Sign. Most of the other characters treat their avatars as just normal gaming devices.
What Yuki Nagato and the remote interfaces for the Data Thought Entity and their rival the Canopy Overmind are in Haruhi Suzumiya.
This is ubiquitous in Ghost in the Shell; it's either remote interface or the person's actual brain in the avatar's body.

Comic Books

The plot of The Surrogates has this happen on a global scale in everyday life.

Film - Live Action

The Matrix does this with the people who plug in.
The titular Avatars in that James Cameron film you may have seen.
This is somewhat of a plot point in the film version of John Carter: He wasn't actually teleported to Mars, just a carbon copy controlled by his consciousness.

Literature

Done via Astral Projection in John Carter of Mars .
Happens a lot in Peter F. Hamilton's The Night's Dawn Trilogy. Ione Saldana and a couple of other characters download their own personalities into bio-engineered Super Soldiers to fight wars. Ione lives in a space station which she herself has never left, and the other characters are often dead. Not to mention Edenists' downloading of personality into habitat 'multiplicities', preserving their memories for centuries after their bodies are dead, or the general effect of possession. Okay, fine. The entire yarn wouldn't exist without this trope.
Tad Willaims Otherland books are set in an AI world, so most of the main characters are avatars of assorted real life people. Orlando Gardiner, in particular, has progeria, so sometimes considers his virtual life more real than his home life. Eventually, his body dies, and he becomes a permenant inhabitant of the virtual world.
In David Weber's Safehold series, Nimue Alban's fabulously wealthy father bought her a fabulously expensive PICA—a robotic body capable of temporarily hosting a human consciousness. Nimue mostly used it for extreme sports, but it was...fully functional...If You Know What I Mean. Merlin explores that aspect later, If You Know Wha—you get the idea.

Manwha

Id_Entity is the manwha version of .//Hack, except it comes with a mode where you can actually play as you sleep, through some kind of poorly explained science babble about the subconscious mind.

Tabletop Games

Role-Playing Games that explore the theme of Transhumanism usually have this as an option (sometimes the only option) for Player Characters. For example, some GURPS settings as well as Eclipse Phase.

Television - Live Action

The heroes of Stargate Atlantis encounter a good Replicator living on Earth. Although she helps them she is too dangerous to remain free, so they place her in a realistic virtual world where she may live as she pleases.