I understand that anime is japanese cartoons, but I don't know if there is any more accurate definition of it.
One of the animes that I have liked the most it's Samurai X, or Rurouni Kenshin in Japanese, It was themed in the late 19th century, and the beginning of the 20th, those were the last days of the samurais, and it was a period for changes in Japan, in which they began to have more contact with occident. I like it because it portraits very well a time period that sounds interesting, the clash of two worlds pretty distant between them, in one side you have all the way of the samurai, and the traditions associated to them, and in the other you have all the "progress". The anime it's focused on the life of a former assassin, Kenshin, that chooses another kind of life, away from all the violence, so he stays in the house of a woman named Kaoru, after he helps her defending his swordsmanship school. In there he will meet new friends, as his old enemies cross paths with him, but also there will be new ones that will challenge him.
What I like from anime is the aspects from their culture that they put in them, you can see various elements like samurais, or magical creatures from their myths, as you can see normal people in a certain time periods that are really trustworthy, and transport you to that time.
Anime hasn't influenced me in anyway, but I found curious that after watching any anime, some people become pretty fascinated with the Japanese culture and begin to listening their music, dressing like them, or dressing up like the characters from anime series. I haven't bought any original soundtrack from animes, but sometimes I listen in youtube some of the songs from animes that I've watched.
The first anime that I remember was Dragon Ball Z, I also remember that I enjoyed it a lot. The story was about the adventures of a group of martial arts fighters from all around the earth, and even other planets, in fact, the main character was Goku, an extraterrestrial man from an outer space warrior race that was pretty eccentric and loved to eat lots and lots of food, but it was pretty humble.