the "ascii cats" you might see here and there mainly originate from an old japanese message board site called 2channel
while "ascii art" is an umbrella term for pretty much any kind of text-based digital art, the character encoding these cats are made with is specifically shift_jis (a branch(?) of ascii, made for japanese characters). also, the font used for 2channel is ms pgothic (the font you're viewing this site in!)
many of the cat characters widely used on 2channel have names of their own!
people made (and probably still make) fanart and even their own stories using these characters
∧＿∧
( ´∀｀) monā (モナー)
pron. "moh-naa"
from the phrase "オマエモナー" (omae monā - you too!)
links
wikipedia (JP)
knowyourmeme (EN)
niconicopedia (JP)
pixiv encyclopedia (JP)
∧＿∧
( ・∀・) moralā (モララー)
pron. "moh-raw-raa"
from a combination of "moral" and monā's name
links
niconicopedia (JP)
pixiv encyclopedia (JP)
∧∧
(,,ﾟДﾟ) giko (ギコ)
pron. "gee-koh"
from a message board user's weird laughing style ("ギコハハハ" - gikohahaha!)
links
niconicopedia (JP)
pixiv encyclopedia (JP)
∧ ∧
(*ﾟーﾟ) shii (しぃ)
pron. "shee-ee"
from the original creator's username
links
niconicopedia (JP)
pixiv encyclopedia (JP)