"I am not a student."
It's a difference of nouns vs adjectives. I'll try to explain :-)
是 is used to say someone is a noun. (Examples: I am a student. You are a teacher. We are people.) When we make a negative, it becomes 不是. (Ex. I am not a student.)
- 我是学生 = I am a student (noun)
- 我不是学生 = I am not a student (noun)
In your other example, it is describing someone as an adjective. (Examples: I am happy. You are fast. We are nice.) In the positive, it would be 很 instead of 是. (我很高兴). Negative sentences however, do not use 不很, just 不.
- 我很高兴 = I am happy (adjective)
- 我不高兴 = I am not happy (adjective)