As far as I know, an acid is something that gives off a proton or hydrogen ion $H^{+}$.
But when I look at Citric Acid,
There are three $COO^{-}$, which I think is a carboxyl group, that lacks $H^{+}$.
They seem to rather demand a hydrogen ion from its environment than donate one, which make it basic and not acidic. But the name is still citrate or citric acid. Can someone explain?
It depends on the pH of the solution. At higher pHs the carboxyl groups will be dissociated (as drawn). At lower pHs the equilibrium will shift and most of the COO- will become protonated.
In general, carboxylic acids are relatively weak compared to stronger acids that dissociate almost completely when added to an aqueous solution. For example, acetic acid, or vinegar, will not burn a hole in cotton clothing, whereas sulfuric acid (the kind in an automobile battery) will easily burn a hole in cotton cloth.
Also, don’t confuse carboxylic acids with their carboxylate salts. Sodium citrate, where the three protons are replaced with Na+, will act as a base when added to water, and raise the pH.
The crux of the matter is that in biochemistry the names of conjugate acid-base pairs (in this case, citric acid and citrate) are used interchangeably. Thus, the picture you have shows citrate (which, as you correctly say, is not an acid), but books may still refer to it as citric acid. See also: Why are lactate and lactic acid used synonymously in biochemistry?
There is a nomenclature confusion. The word citrate means 'a citric acid who already donated protons'. The reason why your books shows the picture with COO- and not COOH is that because at physiological pH the citric acid immediately gives away protons.
The same things applies for amminoacids, who can both be written with NH4+ and COO- groups OR you can use NH3 and COOH. It means the same thing, just a different molecular state.
