Question

What do abbreviations after a word in the dictionary tell you about the word

Answer 1

The abbreviation after a word in a dictionary tells you what part of speech the defined word is.
Ex: (vb) is verb ; (n) is noun ; (adj) is adjective.

Answer 2

If you're referring to the things like adj. or adv., then they tell you what type of word it is. For example, adj. is short for adjective, and adv. is short for adverb. :)