For this, you will obviously need to be in a Syndicate. A contract is when a player is further along in a challenge than you. You can complete this contract by exceeding them in that challenge (like getting more liquid kills, healing more syndicate members, etc). On the stats and challenges screen, your contracts will be on the right side. You will want to join a Syndicate that has players with the same amount of progress as you do, otherwise, if you join one with members that have been playing for a while, your contracts will be for ridiculous values.