Mission running is PvE content: getting missions from NPC agents and then completing them for the mission rewards and the bounties from any hostile NPCs you destroy along the way.
If you decide to pursue mission running as a source of income, you’ll
want to build up standing with a particular NPC corporation in order
to get access to higher tiers of mission agents (which can then give
you harder missions with better rewards). Training the skills Social
and Connections will grant you quicker access to higher-level missions.
In order to tackle these harder missions you’ll also want to continue upgrading to larger classes of ships such as cruisers and battlecruisers. In general, you’ll want:
Under the old alpha clone rules the only battlecruiser that alpha clones could use was the Gnosis. however, since December 2017 (EVE Vegas) alpha clones can now fly up to a battleship.
You can read more on the UniWiki ‘Missions’ page.
Once you’ve received a mission from an agent, you can look it up in the EVE-Survival mission listings to get more details about what kind of dps you might face from enemies in the mission and which enemies may trigger more spawns.
Killing ships of the four major empires will often net you large penalties to faction standings with them. If those standings get very low (below -5.0), you will be shot on sight in their space. For that reason, it is recommended to decline missions that send you to fight the four major empires (Amarr, Caldari, Gallente, and Minmatar).
You can see the icon for the type of enemies the mission will send you to fight in the Mission Objectives. You can decline one mission per four hours from a given agent without penalty, and receive a new one in its place; declining frequently will carry a standings penalty with the mission issuer. If you need to decline more frequently, consider finding another nearby agent and doing missions for them while you wait.
Mission running produces a fairly steady stream of ISK - the mission rewards will pay out a certain amount, plus you’ll also be paid the bounties for any hostile NPCs you killed.
You’ll also be rewarded with loyalty points (LP) for the NPC corporation that is giving you the missions. These LP can be traded for items from the NPC corp’s Loyalty Point Store.