[subject] + 很 + [adjective]
很 goes in front of an adjective that is the predicate. Its dictionary meaning is "very", but here it usually adds little: the adjective needs something before it, and 很 is what goes there.
Translating 我很忙 as "I am very busy" overstates it. Without 很, a bare adjective sounds like half of a comparison: 今天冷 suggests "today is cold, unlike some other day".
English puts "is" before an adjective. In Chinese that slot takes 很; 是 links a subject to a noun, not to an adjective.
In speech, a stressed hěn carries the full degree. Writing can't show stress, so to mean "very" on the page, use 非常 or 特别.
不 replaces 很 rather than joining it. "Not very" is 不太, and 很不 means "very un-".
In a 比 comparison 很 drops out.
很 also goes before a handful of verbs of feeling and wanting – 喜欢, 想, 爱, 希望 – and before 有 in phrases like 很有意思. Ordinary action verbs refuse it: there is no 很吃 or 很走.