What does this mean?
 

What does de arimasu mean? I watch shakugan no shana and one of the characters at the end of just about every sentence says de arimasu.

"de aru" is a more formal way of saying "desu"..."de arimasu" would sound even more formal.

What does de aru mean.

In this case, it means nothing. It has no meaning independent from something that is added to the end of the character's sentences to make them sound more formal. It's pretty silly, and no normal person actually talks like that, nor did anyone in the past, so far as I know. It's an invention.

But in normal Japanese, "de aru"/"desu" is a copula. That is, it serves the purpose of the verb "to be" in English; it establishes equivalences between nouns, adjectives, and noun phrases.

Like this sentence:

Japanese:
Maiku wa gakusei desu.
マイクは学生です。

Translation:
Mike is a student.

gakusei means student, and the "desu" establishes that the topic of the sentence, Mike, is a student.

This is suuuuper-basic Japanese. If you'd like to learn more about grammar, I'm sure lots of people here could help you start to learn the language. Take a look at some of the stickied threads.