Read after the cut for a race-by-race break-down of EVE's most popular and effective mission-running ships, complete with links to some further reading and popular mission fittings.
Caldari ship designs:
The Caldari are the undisputed kings of mission-running. Caldari ships are natural shield tankers, with some being better off with an active shield tank and some able to fit an impressive passive setup. The main Caldari mission-running ships are the Kestrel, Cormorant and Caracal for level 1 and 2 missions, the Drake for level 3 missions and the Raven, Drake, Golem or Nighthawk for level 4. The Raven and its tech II equivalent, the Golem, are the most popular choices for level 4 missions. They can be furnished with a massive active shield tank and deal considerable damage to targets over 100Km away using cruise missiles. Unlike some Caldari ships, the Raven doesn't get a bonus that's restricted to just kinetic missiles. This lets you choose what damage type you wish to deal with extreme ease, making the Raven a great choice for mission-running.
An active shield tank's heavy capacitor requirements can be temporarily satiated by using a Heavy Capacitor Booster and using the shield booster sparingly in pulses when required. Alternatively, a "permatank" setup featuring a very high capacitor recharge rate can be used to keep the shield booster running permanently. A permatank is usually only possible to implement at the expense of damage and the maximum damage per second your ship can tank. The alternative is to run a passive shield setup based on having high resistances, lots of hitpoints and a very fast shield recharge rate. The Drake and Nighthawk are both exceptional examples of ships which can be passive shield tanked.
Amarr ship designs:
If there's one thing Amarr do well, it's outright tank and damage. There are no fancy tricks here, just massive lasers and sturdy armour tanks. Lasers are restricted to EM and thermal damage but many Amarr ships have a large drone bay which can be used to add another damage type into the mix. The best Amarr ships for level 1 and 2 missions are the Punisher, Coercer and Maller. The Prophecy, Harbinger or Zealot are best for level 3 missions and the Apocalypse, Armageddon, Abaddon or Absolution for level 4.
While the restriction of damage types on lasers to EM and thermal may seem to make Amarr ships poor mission-runners, that statement couldn't be further from the truth. Sansha and Blood Raider NPCs are actually weakest to EM and thermal, making Amarr ships perfectly suited to missions against them. Beam lasers have an extreme range advantage that can be used to outrange NPCs and reduce incoming damage, while pulse lasers deal a large amount of damage, have great tracking speed and have a very respectable optimal range. Tech 2 pulse lasers can also benefit from using "Scorch" crystals, advanced long range crystals that can allow pulse lasers to hit from ranges of almost 20km for small guns, 40km for mediums and over 60km for large.