We've looked at the basic types of anime out there, but now it's time to look at the specific genres. Of course, I assume everyone knows what "fantasy" and "romance" are, but what about those genres that don't really exist in the West outside of anime and manga -- or those that don't even use English words?
"Mecha," coming from the word "mechanical", describes any anime or manga largely focused on one distinct aspect: pilotable robots. Sometimes they are gigantic like in Gundam. Sometimes they are the same size as an Iron Man suit as in Bubble Gum Crisis. Sometimes they have a humanoid shape and other times they don't. Sometimes they transform into other machines like planes or even combine from several smaller machines into one giant "super robot". As long as the plot or setting has a pilotable robot at its center, the anime falls into the mecha genre.
If you have ever seen Sailor Moon, you know basically what to expect from a "magical girl" anime or manga: innocent young girls using a magical object to transform into a superhero version of themselves. These girls then fight an evil force -- or sometimes "evil" itself -- with their magical powers. In general, magical girl stories have cute girls that are either prepubescent or on the cusp of womanhood. They tend to deal with the themes of growing up, making friends, learning life lessons, and falling in love. Of course, there are also those occasional anime like Madoka Magica or Kill la Kill that take the standard magical girl setup and turn it completely on its head.
As you can probably guess by the name, these anime tend to focus on characters playing a collectible card game. Almost always, these anime are tie-ins to real-world collectible card games and serve as an advertisement as much as anything else. Of course, rarely are the card games seen in these anime and manga simple card games as we think of them. Often, there is a supernatural or sci-fi element that threatens people's lives or even puts the world itself at stake in the card game. In the West, the most well-known franchise of this type is Yu-Gi-Oh!.
Works in the "monster battle" genre tend to center around characters who train monsters and use them to battle other monsters. Sometimes, monster battles are a professional sport (e.g. Pokémon). Other times, the monsters are trained and battle as part of an adventure or quest (e.g. Digimon). Like card battle anime, these anime and manga tend to be aimed at young boys and usually have building a strong collection of monsters for battle as one of the main forces driving the plot.
In the West, you may be a singer, movie star, TV personality, model, or dancer; but few people are all at once. In Japan, however, there is an entire culture built around such people. They are known as "idols". Idol manga and anime follow people in this industry -- usually starting at their humble roots and following their lives through to super stardom. Many of these are also slice-of-life stories, but others, like the Macross franchise, pull other genres into the mix.
"Harem" is a sub-genre of romance in which there is one male character -- the protagonist -- at the center of a group of female characters who are all vying for his romantic affections. Usually, the females in a harem anime are given distinguishing character traits to set them apart or they simply embody one of the four common female character archetypes. However, it should be noted that having a love triangle or a female-heavy cast does not necessarily make for a harem anime. Only if three or more female characters are romantically interested in the lead male -- and that relationship is the core of the plot -- does it fall into the harem genre. A "reverse harem" is the opposite of the standard harem where there is a single girl being romantically pursued by a group of men.
"Yaoi" is the genre for an anime with a male homosexual relationship at its center. The term comes from the Japanese phrase "yama nashi, ochi nashi, imi nashi" or "no climax, no resolution, no meaning." In Japanese, however, this genre is also called "BL" from the English words "boys love" -- which explains the genre as a whole perfectly. Western anime fans will often use the word yaoi to refer to homosexual pornographic content; however, the term does not necessitate that the work be pornographic in nature.
However, when wanting to expressly state that an anime or manga -- while having a male homosexual relationship at its core -- is not pornographic, the genre term "shonen ai" (literally meaning "young boy love") is usually used.
The female counterpart to yaoi is "yuri" which describes any anime built around a lesbian relationship. "Yuri" literally means "lilly" in Japanese and is used as a metaphor for women. Like with yaoi, yuri is often used to refer to animated/illustrated lesbian pornography though non-pornographic works are also technically yuri.
As shonen ai is the term for non-sexually explicit yaoi among Western anime fans, "shoujo ai" (literally "girls love") is the term for non-sexually explicit yuri. Interestingly enough, however, shoujo ai refers to something very different in the original Japanese: those fans who love Loli-style characters.
In English, "hentai" is the genre of hardcore anime/manga pornography. In Japanese, the word simply means "perverted" -- although not always in the sexual sense -- and does not refer to manga and anime specifically. Hentai can be anything from a simple innocent romance to tentacle rape with everything in between. All that it needs is explicit and graphic sexual content to fall into this genre.
As hardcore is to softcore, hentai is to "ecchi". A work falls into the ecchi genre when the sexual content goes beyond simple fanservice but not so far as graphic sex. Ecchi works tend to focus on the sexual content over (or at least on an equal level to) the plot and comes complete with groping, revealing outfits, gratuitous boob bouncing, and often full frontal nudity -- all for the sake of simple titillation. The word "ecchi" is actually just the Japanese pronunciation of "H" -- i.e., the first letter in "hentai" -- and like hentai, it isn't defined as a genre in Japanese but just another word for "perverted".