I consider myself an anime connoisseur. I've sampled a lot of anime, but I'm picky about ones I actually like. I rate most anime I see as a 5 if they are just average and nothing special. I usually watch an anime for about four episodes to see if it looks like something worth continuing to watch. If it doesn't, then I drop it.
For an anime to be objectively good, it should have the following qualities that brought me to love the first anime I ever saw (Sailor Moon):
good art (realistic and beautiful)
excitement (has action)
music (lyrical songs in the show besides the opening and ending themes)
plot (a storyline that is at least a little unique and not episodic in nature; character motivations are explained)
romance (the better ones are not just based on appearance)
sadness (makes me cry at least once, which isn't that hard)
For an anime to be subjectively good (in other words, for me to like it), it should satisfy these requirements:
contains a moral message
real world significance - even if it's a fantasy, themes in the anime must be able to translate to the real world
depth of character - where characters are not just shallow stereotypes of good or evil, but have reasons behind what they do
cynical/dark - shows how evil the world and some people can be
This site doesn't let me show all my favorites on the left hand side of my profile, so here are my full lists:
Favorite anime: Bishoujo Senshi Sailor Moon, Blood +, Boogiepop Phantom, Code Geass, Death Note, Earth Maiden Arjuna, Elfen Lied, Eureka Seven, Fushigi Yuugi, Inuyasha, Kino no Tabi, Kodomo no Omocha, Madlax, Mahou Shoujo Madoka Magica, Melancholy of Haruhi Suzumiya, Monster, Nana Seven of Seven, Nanaka 6/17, Noein, Now and Then, Here and There, Please Save My Earth, Serial Experiments: Lain, Shadow Star Narutaru, Shugo Chara, Speed Grapher, Tenkuu no Escaflowne, Wallflower
Favorite anime movie: Full Metal Alchemist, The Girl Who Leapt Through Time, Tokyo Godfathers
Favorite anime character: C.C., Haruhi Suzumiya, Hiroko Kaizuka, Kino, Lain Iwakura, Lalaru, Lelouch, Light Yagami, Lucy/Nyu, Miaka Yuuki, Miyu Yamano, Nanaka Kirisato, Naozumi Sudo, Rei Ogami, Sana Kurata, Sunako Nakahara, Teru Mikami
Favorite anime character from Sailor Moon: Heavy Metal Papillon, Hotaru Tomoe, Souichi Tomoe, Usagi Tsukino
Favorite anime genre: fantasy (not sci-fi) & shoujo.
Least favorite anime genre: mecha because robot fights bore me. I'd rather watch hand-to-hand combat, sword fights, or magic fights anyday. There are only a few mecha anime I actually like a lot, and I like them because their robots don't fight much.