不 and 没 are both adverbs, which can be followed by verbs or adjectives.
没 negates objective facts and is mostly used in the past and present tenses.
她昨天没来学校。（in the past)
She didn’t come to school yesterday.
这个手机没坏。(in the present)
This cell phone is not broken.
不 negates subjective will and feelings and can be used in the past, present and future tenses.
By the way, 不 is often used to negate a habit.
他吃素，不吃肉。He is a vegetarian, so he doesn’t eat meat.
他不抽烟，不喝酒。He doesn’t smoke or drink.
没 only goes with a limited number of modal verbs and psych verbs, such as 能，敢，想 and so on. It’s always used in past events.
我没能帮他。I couldn’t help him.
我没敢进去。I didn’t dare to go in.
Wǒ méi xiǎng bāng tā.
我没想帮他。I didn’t want to help him.
不 can go with all modal verbs and psych verbs. It can be used in the past, present and future events.
她不会开车。She can’t drive.
我不喜欢看电视。I don’t like watching TV.
Nǐ bù yīng gāi hē tài duō.
你不应该喝太多。You shouldn’t drink too much.
To negate the verb 有，you can only use 没; to negate the verb 是，you can only use 不.
我没有哥哥。I don’t have a brother.
房子里没有人。There is no one in this house.
Wǒ bú shì xué sheng.
我不是学生。I‘m not a student.
Tā bú shì Měi guó rén.
他不是美国人。He is not American.
没 can only be used with a few adjectives to negate the change of a situation.
She is not well yet.
Long time no see. You look the same as before.(neither gain nor lose weight.)
不 can be used with most adjectives to negate the state of somebody or something.
他的弟弟不高。His brother is not tall.
这个包不好看。This bag is not nice.
没 can be followed by a noun, to indicate 没有. But 不 can NOT be followed by a noun.
他没钱买新手机。He has no money to buy a new cell phone.
咖啡店里没酒，只有咖啡。There’s no wine in the coffee shop but coffee.
没关系 and 不客气 are fixed expressions. They are used in response to apologies and thanks respectively.
Méi guān xi.
Bú kè qi.