Crossovers are usually 1 work crossed with another, normally done by a co-op initiated by the creators of both works
example of Crossover
High School DxD x The Ambition of Oda Nobuna
Hyperdimensional Neptunia x Seiran Kagura
Dragon Ball x One Piece
Homage is a work paying respect to another work or multiple other works though they usually have permission to do so
example of Homage
Tensura homage of Dragon Ball Z and Street Fighter (Veldora using Super Saiyan Transformation and Kamahameha, as well as several Street fighter moves)
Arifuruta homage to High School DxD and Certain Magical Index (Tio's character concept is a homage to Orphis from DxD, Shea's Hammer is a Homage to RWBY Nora Valkyrie's Hammer)
High School DxD homage to Gundum and Dragon Ball (Issei's first move used is a reference to the Kamahameha, Sirzechs and Milicas' name is a mix of character names from Gundum Wing character Zeck Marque and Milacrado Peacecraft)
Fanfiction is works done by fans for series they love
works who have high fanfiction volumes
Naruto, One Piece, Bleach