Note that, even more so than in most forms of media, the language used in anime is heavily stylized and quite different from the way actual Japanese people speak in real life. あたい Originally used by women in certain red light districts, it later spread out and was eventually picked up by those wanting to cultivate a "bad girl" image.
Take everything you hear in anime with a grain of salt — foreigners who watched too much anime while learning Japanese are easy to spot. Characters who use this pronoun are implied to be lower-class, uneducated, and flippant.
Still pops up as a pronoun once in a while, typically by the military sort who might refer to himself as そなた/其方 An archaic form.
Each of them makes a different statement about the speaker's gender, age, social status, and relationship with the addressee(s).
Don't confuse it with 儂 In popular media it's reserved for elderly men only (except for some Jidai Geki dramas and suchlike), but in real life it used to be popular with men of all ages, especially in the mid-western regions of Japan.
By now its usage has faded among the younger generations, due to the effects of the aforementioned popular media.
To capture a little of the flavor, English translations sometimes use "this (category of person)" — this little girl, this humble peasant, this Badass. The "wrong" pronoun can be a moment for comedy (see Different for Girls); the specific choice can say a lot about the character speaking.
And it's easy to avoid revealing characters' names, for whatever reason.
私, わたくし An ultra-formal term, often used in anime by characters who are profusely polite, very sophisticated, or somewhat old-fashioned.