Third Wheel

Any instance where at least one other character is hanging around a couple and some form a social tick is getting in the way. He/she can either be

A) a nuisance to the couple and they consistently tell him/her to leave them alone,

B) a character who doesn't want to spend time with the couple but they drag him/her along, or

C) a character who is pulled into being with a couple due to circumstance and no parties necessarily have ill will toward each other, but awkwardness may ensue anyway.

Usually does not translate into a relationship between a couple and a third party who genuinely like to spend time together in a friendly way, since that would imply little to no awkwardness.

Examples of Third Wheel include:

Anime and Manga

There's a Type A example in the Boys Love manga Challengers. The foreign transfer student Rick quickly becomes a nuisance after he starts following the main couple around in the hope of getting with Uke and Horrible Judge of Character Tomoe before Kurokawa, who has more morals when it comes to relationships, upgrades their relationship from Just Friends to Official Couple. Even after they become lovers he still hang round with them just in case.

Film

In The Adventures of Milo and Otis, main characters Milo the cat and Otis the dog spend most of the film trying to reunite, but when Milo falls in love with another cat, Otis becomes this for a little while. Otis is mostly a case of type C.
Both Sherlock Holmes films are ripe with this trope. Your Mileage May Vary on whether Holmes is a third wheel to Watson and Mary, or if Mary is a third wheel to Holmes and Watson; Heterosexual Life Partners or Ho Yay, your choice.

Western Animation

Timon and Pumbaa of The Lion King certainly become third wheels once Simba and Nala meet up again. It definitely is mostly type C, but Simba kind of politely lets Timon know that the two of them aren't wanted, making it somewhat type A.
In "Dan Vs. The Fancy Restaurant" Dan accompanies Chris and Elise on their anniversary dinner.

Dan: Why is having a third wheel a bad thing? Third wheels provide stability, that's just physics.