In anime, character eyes usually have highlights that are usually white. They seem like a light reflection or something.
Do the highlights have a name? What are they supposed to be?
For illustrative purposes, I have taken the first four images off a Google search for
eyes (note: you may not want to perform this search yourself, as there is at least one disturbing image on the front page).
As you can see, all of these images contain a similar "white space" / highlight. That's because if you're taking a photo, you need a light source - and the eyes, being reflective, are going to reflect that light source.
These "white spaces" are called catchlights, and are naturally found in film and in photography. It is unsurprising that animators and artists would imitate this for added realism - Disney's been doing it since forever.
Of course, catchlights are somewhat more pronounced in anime simply because characters drawn in an "anime style" tend to have large eyes - and hence, correspondingly large catchlights.
For more information about catchlights in photography, see the question In portrait photography, what is a “catchlight”? at Photography.SE.
The Japanese term for "specular highlight" or "catchlight" is 「鏡面ハイライト」 (kyoumen hairaito), meaning "mirrored surface highlight" or "specular surface highlight." 「鏡」 can be pronounced "kagami," "kyou," or "kei" and means "mirror,"「面」 is pronounced "men," "omote," or "tsura" and means "mask" (as in the word 「仮面」[kamen]), "face," or "surface."
The sort that appear in human eyes are called 「人物のハイライト」 ("jinbutsu no hairaito" = person's highlight), 「眼球のハイライト」 ("gankyuu no hairaito" = eyeball highlight) or 「眼球ハイライト」 ("gankyuu hairaito" = eyeball highlight). A specifically manga-style specular highlight can be referred to as 「漫画的に描いたハイライト」 ("manga-teki ni kaita hairaito" = highlight drawn in manga-style).