1. Like many of you guys mentioned, anime is so varied that it's never boring. So many types of plots and genres are depicted by anime that it hooks the viewer. From cooking, to family life, to school life, to action-pace, to mecha series, to samurai series, to sci-fi/futuristic series, to lovey-dovey series, to *harem* series, and on.
2. Anime ENDS. Like, whoa?! Many other series out there don't freaking end. (Asian soap operas, American cartoons, reality TV shows, etc.) Naruto, Detective Conan, the Gundam franchise, and some other series are a whole other story.
3. A branch-off of reason 1, the plots of anime are just simply awesome. They integrate so many genres and factors or isolate a particular genre/factor and they create great storylines that don't lose the audience.
4. Anime characterizations, exaggerations, and mannerisms are things that make anime so unique and memorable. Some quotes become stuck and live on; for me: "Makenaide!" "Ku-chon ku-chon no ke-chon ke-chon ni shite kureruwa." "Ore wa tatakaou." and the best, "Dare da omoute igeru!" from TTGL.
5. The art: the art is so unique and it's so beautifully drawn that it sucks in the viewer with just it's art alone. Take a look at Gundam designs and Miyazaki's beautiful scenery and wonderful creatures; take a look at the many cute, sexy, and strong characters; take a look at the complex or seemingly simple architecture design that graces the everyday school to a complex metropolis in 3000AD; that's what I mean.
6. The music: Anime is characterized by the characters and plots right? I think the other factor that people gloss over is the music. The OSTs from various animes are ridiculously ingenious, like the composers Yoko Kanno (Vision of Escaflowne) and Yuji Kajiura (Erementar Gerad, Tsubasa Reservoir Chronicles). On top of the OSTs, the opening songs and the ending songs catch the viewer like nothing else, ranging from rock to pop to trance to choral. OP/ED does it like this: careful presentation (selected pictures, slow or break-neck speed or both) with the right music will catch the viewer, hook, line, and sinker.
7. The language spoken: The seiyuu industry, if nothing else, is nothing short of amazing. The voice talent of many people make the anime alive. With the right emotions, the suspended reality becomes so real, whether it be someone crying or someone yelling, it gets at you. On top of that, the Japanese language is just so beautiful in itself; I just love hearing it from modern day Japanese to ancient flowery Japanese.
8. This is just me, but anime creates a line of products, such as Gundams or figurines. Without the anime, there would be no toys or models. (Ahh, my gundam collection! Don't go!)
9. Anime, at it's very core, is a journey, and better yet, a hero’s journey. Now, I'm not too sure if people here are familiar with the literature concept, but it's prevalent in every story ever told. All anime does is transfer that hero’s journey onto the screen with every other factor in: the unique art, the music, the flow of frames, the voice acting, etc. etc.
So what is a hero’s journey? It entails a hero or heroine that is catapulted into an adventure, willingly or unwillingly. Rosette of Chrono Crusade willingly goes out to search for her brother, Joseph, while trying to save the world from apocalypse. Kira of Gundam SEED is unwillingly put into a war between Naturals and Coordinators while trying to find the middle line. The adventure often has a goal or purpose, if you will, and the hero/ine/es then embark upon the adventure to resolve the conflict, reach the goal, end the quest. Erementar Gerad is perhaps the clearest example where Coud is thrust into an adventure to reach the Garden of Eden while trying to fight off bad guys and come to understand and love Ren, his Eden raid.