Distribution missions are missions given by distribution agents. They are usually courier missions in which you have to transport some cargo for the agent. An exception to this are some low-level mission chains (A New Venture, for example) which do involve combat. It is recommended having a small combat-capable ship (Frigate/Destroyer) at or near your mission base.
A courier mission is a mission to either collect a piece of cargo from one station and deliver it to another or to collect a piece of cargo from a deadspace location. When a courier mission is accepted, the necessary cargo is spawned in your personal hangar at the "Pickup Location". You then need to haul it to the "Drop-off Location" using your ship. Usually, the agent location is either the pickup or drop-off location. Once docked at the destination station you can 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.
TIP: Although located in your ship inventory, cargo held in containers will not be accepted.
Aside from any mission-related hazards you only have to deal with the usual hazards of travel (Gatecamps, suicide gankers, Warp Interdiction on Stargates in Null security space, and so on). Level 1 missions will keep you within the agent's constellation, level 2 and level 3 will possibly send you to an adjacent constellation, level 4 missions will always send you to an adjacent 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 issuing agent's station, which is usually the pickup station.
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 distribution missions you will need a hauler as cargo sizes will be in the 4,000–8,000 m3 range. Note that it is beneficial to have more cargo room than required as you can then run missions for multiple agents at once.
Level 1 and level 2
The Sunesis is a great option for those levels. It has a base cargo capacity of 600 m3 which can be increased by expanded cargoholds. You can fit it to align in under two seconds even if your skills are not especially good. This means most gatecamps will struggle to lock you fast enough and you can comfortably use it in highsec and lowsec. When you fit a bit of a shield tank you also have enough EHP that it's very difficult to kill you with a smartbomb camp. As rigs you usually choose hyperspatials to warp faster and thus reduce travel time.
If the Sunesis is too expensive for you, you can also use a T1 exploration frigate (Magnate, Heron, Imicus, Probe). They have a base cargo capacity of 400 m3 so it is suggested to fit at least one expanded cargohold. You can achieve align times of under three seconds with fitting and good skills but you cannot go under two seconds (at least not reasonably). This means gate camps might more often catch you. If you're not wardecced, this isn't a big issue in highsec. In lowsec, you might want to use MWD + cloak trick (which is also an option for you in the Sunesis).
If you have already access to a Blockade Runner, you could use it not only for higher level distribution missions but also for the low levels.
Level 3 and level 4
As the required capacity can range up to 4,200m3 for level 3 and 8,000m3 for level 4, you will need to move up from a frigate to a bigger ship. A good option which is cheap and has low skill requirements are T1 fast transports (Sigil, Badger, Nereus, Wreathe) which are all able to reach the required cargo capacity with sufficient expanded cargohold even at low skill. While you're going to need to be able to fit up to 8,000 m3, most missions require 4,000-5,000 m3 which will allow fitting more inertial stabilizers for a faster align speed. Hyperspatial rigs are used for higher warp speed. Cloak and MWD are nice to have for the MWD + cloak trick. Anything that increases warp speed, increases agility, or decreases inertial modifiers will help speed up your distribution runs. While there are some differences between the different fast transports, you mostly want to go for the one where you have the best skills.
Blockade Runners are arguably the best option for distribution missions all around. All of them can carry more than 8k m3 if you fill the lows with expanded cargoholds. Their align time is generally decent and you can switch lows for inertia stabs if you don't need the full cargo capacity. They warp fast with a base speed of 6 AU/s which goes up to 7.5 AU/s with skills and even 11.4 AU/s with T2 hyperspatial rigs which are usually used. A covert cloak increases the safety and you can even fit an interdiction nullifier to avoid bubbles in nullsec. All blockade runners are decent options. The Prowler and Viator might be a little bit better but if you have committed to another T2 option, rather just stick with it.
Warning: Blockade Runners are considered high value targets. As they can't be cargo scanned most gankers will assume they have valuable cargo and gank them whenever they catch them. Be very careful running distribution missions in Blockade Runners around populated areas.
Distribution missions are fast to complete and thus you can quickly finish 16 missions to gain a storyline mission. This is especially true if you have multiple agents close together ideally in one system or even station. Storyline missions are especially wanted for faction standings. So distribution missions are a good way to increase those. If this is the best way depends on a lot of factors. You might rather just first focus on corporation standings and just slowly get storyline missions while doing security missions. There are also one time alternatives to gain faction standings such as Career Agents, Circle Missions, and COSMOS. Epic arcs also give faction standings and can be done every three months.