Some examples: Wiz from Konosuba, Touka from Tokyo Ghoul, Sanji from One Piece
Note that while many have a plot reason, I'm referring to those who don't.
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2Perhaps the Peek-a-bang trope might be what your asking about? – Wondercricket Oct 29 '18 at 12:38
It's mostly done because it looks cool. It's a very simple design choice that introduces asymmetry to a character's face, makes them a little more interesting to look at.
In general, hiding part of the face makes a character seem more mysterious or withdrawn, so it's useful to contribute to characterization for those characters that are shy or aloof, it reinforces that characterization.
There's no special significance to the right eye, in fact it's not uncommon for the side of the face that's covered to switch back and forth, either due to mistakes made by the artist/animator or due to the angle the character is seen at, or due to an actual change in the character's design (Sanji, for example, switches which side of his face is covered by hair after the time skip in One Piece).
