Much vs many - what's the rule?

Many goes with countable nouns. Much goes with uncountable nouns.

Countable nouns can be counted individually - they have a singular and plural:
"Many people came." (people: 1 person, 2 people)
"Many mistakes were made."
"How many emails did you get?"

Uncountable nouns can't be individually counted - they're masses or concepts:
"There isn't much time." (time: you can't say "one time, two times" in this sense)
"Much of the work is done."
"How much water do you need?"

Much and many are most natural in negatives and questions. In positive statements, a lot of sounds more natural:
"There are a lot of people here." ✓ (not: There are many people here - sounds formal)
"There isn't much traffic today." ✓
"Are there many seats left?" ✓

"There's too much noise." (uncountable)
"There are too many people." (countable)