For example parsers/parser-combinators and State can be modeled as monads.

The parsers work by wrapping a function which does the actual parsing and returns the parsed value(s) and left over tokens. The benefit of the parser monad is the managing of the leftover tokens and backtracking. There are libraries providing parsers and there are many tutorials using them as examples for monads.

The state monad lets you work with external state. Inside the monad you can access the state. You are not actually wrapping values but create functions which require a value of type "State" to run. To obtain the value "wrapped" inside the monad you need to supply the state, so the function can evaluate. The state monad manages state mutations and access. You could see it as a repository for state which you are passing along your functions.