Hentai and Ecchi
Hentai - literally "perverted," pornography.
Examples: Kite, Cream Lemon, Mezzo Forte, Welcome to the Pia Carrot, Akikan!, Akiba Girls, Ebichu, Koihime, Level C, La Blue Girl, Teacher's Pet
Examples: Elfen Lied, Golden Boy, Green Green, Chobits, Kanokon, Popotan, School Days
Because of the vastness of the Internet, hentai and ecchi are extremely easy to abtain. Because of this ease and availability, it's becom the stereotype that all anime is inappropriate. That is not true. Hentai and Ecchi exist, though, and I advise parents to keep a watch on their internet history. There many popular themes in hentai including "tentacle rape," incest, beastality, torture, and other forms of perversion. After all, that's what Hentai means. Ecchi, while it is not straight out pornography, involves things like big, bouncing breast, revealing clothes, panty shots, and sexual content - usually offscreen, but with audio. There are more ecchi anime available than hentai.
There seems to be a fantasy Japanese girls have about falling in love with their teachers. I'm against teacher-student relationships 100%. Most of the time, there is some convienient excuse like the teacher is fresh out of college or a childhood friend. This plot be treated very casually in anime (such as in Marmalade Boy) or can be used for a hentai plot line, which makes me think they know it's wrong, but will do it anyway.
Examples: Great Teacher Onizuki, Please Teacher, Kodomo no Jikan, I, My, Me Strawberry Eggs
This is anything the creators add to an anime that excites or amuses the audience, but is not necessary to the plot. For example, if we get a glance of a character from another anime, or if the characters say, "You're not in a manga, you know." The problem comes in with sexual fanservice. This includes panty shots, jiggling breast, too-tight clothes. Female characters are usually the objects of fanservice because anime watchers are primarily young men. There is usually a onsen (hot springs) episode where the girls are bathing in the hot springs. Or, they all go to the beach and the girls are in bathing suits; or, for the school culture festival they have a maid cafe.
Examples of anime with large amounts of fanservice: The Melancholy of Haruhi Suzumiya, Tenjho Tenge, Ouran High School Host Club
Moe: literally means "fetish" - refers to anything that fans think is cute or adorable
Examples: Lucky Star, Manabi Straight, Peto-Peto-san
Moe is nessessarily bad. The point when moe causes problems is when it activates fanservice. For example maganekko moe refers to fans who like girls with glasses. It's not wrong to be attracted to something, but to obsess over or use moe to cause arousal is wrong. Every year in Japan, fans meet online to vote for the most moe character of the year. This is the Anime Saimoe Tournament. Last year's winner was Kagami Hiiragi from Lucky Star.