The newspaper this allows a villain to pick up randomly generated missions in the current zone. Once you've done a few of these missions, you can do a Mayhem where you earn additional badges and temporary powers.
To get a paper, you have to talk to one of the brokers in a zone that's in your level range. The simplest way to do this is go to Port Oakes when you are between levels 5 and 15 or so. The first time you enter the zone, you'll automatically get a broker as a contact.
Select him from your contact list and follow your map to him and you'll get the Newspaper (more simply known as a paper). The paper will be listed as your first contact in your contacts list for the rest of the game. You can read the paper and select from 3 available random Door Missions.
There's a few standard types of missions (defeat a boss, rescue, retrieve item), but sometimes you'll get "robbery" missions which can be more interesting. I for one hate rescue missions more than just about anything so I'll always avoid those if possible.
One tip: if you get only missions you hate, you can leave the zone and when you come back, there will be three new options. Either that or you can use the Drop feature to get rid of one you want by instantly completing it then immediately get another. But the easiest way is the new feature that lets you abandon missions as often as you want at no penalty (and also no reward of course).
The best part about these missions is that they're always in your zone, you can get them from anywhere (you don't have to run to a contact) and after a certain number you get the option (meaning you don't have to do it if you don't want to) of doing a Mayhem mission.
The negatives are that they don't have any story associated with them so they can become tedious and you can only get paper missions if you are in a zone that matches your current level range that also has a broker in it (no PvP zones).