If you want to run missions, you need to find a high-quality agent who is available to you, and, preferably, near your current location. Read this EVE Online Agent Guide to learn more about finding agents, finding the best agents for you, and how to make good money and standing increases from mission running.
Inside of NPC space stations, you will see a list of agents who have offices in that station. If those agents are available to you, you can double click their portraits to talk to them, and then you can request a mission from them. An agent's level, ranging from 1 to 5, determines how hard the missions from that agent will be (with level 1 being easy, level 5 being hard).
Different types of agents will give you different types of missions, depending on the types. See below for the approximate probabilities of getting certain mission types from certain agent types.
In EVE Online, agent quality is in the range -20 to 20. Agent quality is a measure of how much (or how little) money an agent will give you for the missions that you do for him or her. Level 20 agents are very generous with their ISK rewards, while Level -20 agents are very stingy.
Agents have a level, and it can range from level 1 through level 5. This number is a measure of how hard the missions are that a particular agent will give you. Higher level agents usually give better rewards for missions, as well, including better ISK rewards, standing increases, and items.
Level 1 missions (that is, missions given by level 1 agents) are easy, and level 5 missions are very hard. Tutorial Agents can kind of be considered to be level 0 agents, because their missions are easier than level 1, usually.
A higher-level agent always gives better rewards than a lower-level agent, even if the agent quality of the higher-level agent is worse. For example, if one agent is level 1 with quality 17, while another agent is level 2 with quality -20, the level 2 agent will still give you better rewards than the level 1 agent, even though the level 2 agent quality is much lower.
Use the Nearby Agent Search to find good agents to work for.
Or, in-game, your character sheet is a good starting point for finding the highest-quality agents that are currently available to you. Your available agents depend on your standing with factions and corporations, so look at the standings tab of your character sheet. The best agent available to you is probably going to be an agent from the faction that you have the highest standing with, in the corporation of that faction that you have the highest standing with. Open up the information window for that corporation. Read the next section to find the best agent available to you in that corporation.
Use the Nearby Agent Search to find agents of a particular corporation to work for.
In the corporation window, click the Agents tab to see a list of agents in that corporation. This list will show you which agents of this corporation are available to you, and which are not. The agents list is broken out into the different types of agents in that corporation. The agent list is sorted with lower level, lower quality agents at the top, and higher level, higher quality agents at the bottom, so the very best agent of each type will be at the bottom of the available agents list.
Use the Nearby Agent Search to find nearby agents.
Sometimes the best agents are far away, and you just want to find an agent who is nearby. To do this, open up your map. If there is a specific corporation that you want to try to work for, and you are looking for a nearby agent of that corporation that is available to you, open the information window for that corporation, click the agents tab, then click the button at the bottom that says "Show on Map." This will light up the solar systems that contain agents of that corporation. It will not, however, tell you if any of the agents in those systems are available to you.
Make a note of the nearest solar systems that have agents of the corporation that you want to work for. (You can set a waypoint to each highlighted solar system to find out how many jumps away it is.) Then, go to your map settings, and highlight solar systems based on Your Available Agents. The solar systems that contain agents available to you will glow green. Larger glowing spots mean that there are many agents available to you in those systems. Find the solar systems that you noted before, with the agents of your desired corporation. If that system glows green for you, mouse over it to see a list of available agents in that system. If there is an agent of your desired corporation, make a note of it and check any other nearby systems to find out if there is a better agent in one of those. Then just set a waypoint to the desired solar system and pay a visit to the agent you discovered.
The best agents in EVE Online are the ones that have the highest level and quality of all the agents in the game. There are no level 5 agents with quality 20, but there are level 5 agents with quality 19. See a list of the highest level, highest quality agents in EVE.