From the game's encyclopedia
Bastards are illegitimate children born out of wedlock. As such, they are barred from inheriting Titles. However, their parents can legitimize their child with the approval of their House Head.
Which House a bastard belongs to is determined by the Doctrines of their parents' Faiths when they are born.
If Bastards are granted Titles or have children of their own, they will found a new Dynasty.
The game's text for patch 1.20.1.2. Words in blue are entries of the encyclopedia: hover one to read it, as in the game.
Bastard is an entry of Crusader Kings III's in-game encyclopedia. The game explains it: "Bastards are illegitimate children born out of wedlock." It covers Legitimized Bastard. The game also links it as Bastards and Bastardy. Patch 1.20.1.2, from game/common/game_concepts/00_game_concepts.txt and game/localization/english/game_concepts_l_english.yml.
Source: game/common/game_concepts/00_game_concepts.txt, game/localization/english/game_concepts_l_english.yml and the English localisation, Crusader Kings III 1.20.1.2.