A courier mission is a mission that you receive from an NPC agent where the agent asks you to either bring an item to the agent's station from another station, or the agent asks you to deliver an item to another station. The key things to consider when running a courier mission are whether your ship is fast enough, has enough cargo capacity, and has enough defenses to make it safely from your starting point to your destination and back again.
How to ensure that you have enough cargo capacity for the mission
Level 1 courier missions probably won't require you to have much cargo capacity in your ship to carry the items you need to transport. However, higher level courier missions tend to require a large amount of cargo capacity for you to be able to transport the goods in one trip. Sometimes, if the route that you need to take isn't too long or dangerous, it's OK to make multiple trips, but sometimes that just won't work.
In order to have enough cargo capacity, you might be able to buy an Expanded Cargohold module that will expand your cargo capacity enough to carry all of the items that you need to transport. However, be aware that this will make you more vulnerable, because the extra cargo space reduces your ship's structural hit points. Additionally, it might be the case that there just isn't an Expanded Cargohold module that gives you enough of an increase in cargo capacity for you to be able to transport the goods in one trip.
An alternative option is to buy a larger ship that has adequate cargo space. Think carefully before making such a large investment, however. You want to choose a ship that is fast, but that also has decent defensive capabilities, especially if your route will take you through dangerous areas of space. You should also choose a ship that you will want to use again. It would be unwise to buy a ship that you only really want to use once.
How to ensure that you will be safe as you travel to and from your destination
Sometimes a courier mission can take you through dangerous territory. If you're unsure about whether you can protect yourself against an attack from another player, go into the map view control panel, and set autopilot to choose the safest route from your starting point to your destination, and set the safety penalty slider to the highest number. Your ship will avoid low security space if possible.
If you will have to go through low security space, you should consider having a Microwarpdrive fitted to your ship, or some other propulsion enhancement, to give you a way to run away from an enemy who is trying to attack you. When transporting cargo through low security space, you should be on your guard at all times, and warp away from any dangerous players that appear.
You could also fit weapons to your ship, but generally, since you are on a courier mission, it tends to be best to equip modules that increase your speed, rather than using up your fitting slots on weapons.
How to complete a courier mission
If your agent wanted you to take something from the agent's station to some other station, the way to complete the mission is to click on the mission agent under "Agents of Interest" in the station information window, then tell the agent that you completed the mission.
If your agent wanted you to bring something to the agent's station from another station, you do the same thing. As long as the requested item is in your cargohold or your hangar in the station, the mission will complete once you tell your agent you completed the mission.