What is improper use of a conjunction?

Well, it's technically improper to place a conjunction at the start of a sentence, though this rule is often ignored in casual conversation. For example:

'And he jumped.'

OR

'But he never did it.'

are improper.

Also, every conjunction needs a phrase on either side. For example:

'He threw a fastball and.' is incorrect. You need another phrase, or else people will end up saying 'AND WHAT???"