A Distribution mission is a mission to take a piece of cargo from one station to another station. When a Distribution mission is accepted, the necessary cargo is spawned in your personal hangar at the pickup station. You then need to haul it to the destination ("drop off") station using your ship. Usually the agent location is either the "pick up" or "drop off" location. Once docked at the destination station you may complete the mission by talking to the agent. The cargo only counts as delivered if it is either in your personal hangar at the destination station or in your ship's cargo hold while you are docked at the destination station.
Distribution missions never spawn any hazards of their own; you only have to deal with the normal hazards of Stargate travel (gatecamps, suicide gankers, warp interdiction bubbles on Stargates in NullSec, and so on). Level 1 missions will keep you within the agent's constellation, level 2 and level 3 will possibly send you to a neighboring constellation, and level 4 missions will always send you to a neighboring constellation. 
It is worth noting that although you are at the destination station you can still talk to the original agent remotely to complete the mission without going back to the original station. They will be listed in the station's "Agents" tab after you dock, or you can start a conversation with them using your mission journal. However, until you fly back and dock at their station, they will not give you a new mission.
If a Distribution mission has an item as a reward instead of ISK, then the item will appear in your personal hangar at the agent's station (which may or may not be the dropoff location for the mission).
Level 1 and 2 Distribution missions can be run using frigates, although you may need to use cargo modules in the low slots. Cargo size for L1/L2 missions can be up to 450 m3 in size. For level 3 and 4 courier missions, you will need an industrial hauler because cargo sizes will be in the 4000–8000 m3 range. Further information is in the following section.
For small volume missions T1 exploration ships are best as they have largest cargo holds among frigates. The 450 m3 cargo is easily accomplishable, although in the case of maximum size cargo a Cargohold Extender module may be required. Consider that the Magnate has the most low slots, which may be of the most use for fitting cargo and agility modules, although as the role is not very demanding, even a Heron can do the job in a pinch.
You will likely move up from your Frigate to an Industrial ship. Your goal in hauling is always to move as much as you can as fast as you can, and whichever industrial you choose should be able to handle level II and III missions. Level IV missions can require up to 8,000 m3 of cargo space - keep that in mind as you shop for ships later in your career. Expect to accomplish this in a "fast" industrial (Nereus, Badger, Wreathe, or Sigil) with a decent level of skill and possibly additional modules. These hulls confer the advantage of extra warp speed compared to bulk haulers, along with skill bonuses to align time and base cargo.
Distribution missions are by far the fastest mission type. As a result blitzing distribution missions is the fastest way to receive storyline missions. Even though the distribution missions give low corporation standing gain the storyline mission will give more noticeable corporation standing gain and slow faction standings.