One of the more interesting 1v1 battles out there is a cruiser vs a frigate. From the point of view of an outsider it should be obvious who wins. A cruiser has both the firepower advantage and an advantage of being able to take a lot more damage. However the reality is that these battles are a lot more interesting then you would think. For me the first cruiser vs frigate battle I was in was way back in a DOS game called 'Ancient Art of War at Sea'. In that game sea battles were fought between 1800 era ships of the line, battleships and frigates. Battleships were slow and hard hitting. Frigates were fast and nimble, but not very powerful. Ships of the line were in between. Since these ships could only fire sideways a fast frigate could maneuver so that the larger ship could never hit it. By dashing in an out a frigate could, given a lot of time, eventually take out a larger enemy ship. Eve is similar. Frigates can use their speed and small signature radius to avoid damage from larger ships. However in EVE, the larger ships have some countermeasures. Drones, energy neutralizers being the most important. The most important thing however is to go into a fight with a plan.
Lets look at what weapon systems a cruiser might have that are a threat to a frigate. In order of ease to mitigation we have:
1. Guns: In most cases if the cruiser's guns can hit the frigate with anything close to full damage, the frigate is going to die. The trick for a frigate is to make a cruisers guns useless. This involves using a combination of high angular velocity and a low signature radius so that the cruisers guns can not track the frigate. In most cases this is going to involve getting into point-blank range, paralyzing the target, while orbiting as fast as possible on afterburner. The medium guns on most cruisers just can't hit an afterburner equipped frigate that is orbiting it. The cruiser will typically try to get range and kite the frigate, which breaks the orbit and crashed the angular velocity. Ergo it is important for the frigate to scram (take out the MWD on most cruisers) and if possible webify (further kill the speed) the cruiser so that it is unable to do so. Remember that a Warp Disruptor is pretty useless here as it does not shut down a MWD and most cruisers with a MWD are faster than a frigate with an afterburner. If the frigate fields a MWD as well, then it suffers from a sig radius bloom effectively making the cruiser's guns track 5x as fast. The exception to this is an interceptor. An inty may want to use it's high MWD speed and relatively low signature radius to mitigate damage.
2. Small Drones: Most cruisers have largish drone bays stuffed with light drones. It is pretty typical to face 5x Warrior II or 5x ECM drones. Either case is a big problem. #1 priority for a frigate is to start dealing with those drones once they are launched. The good news is that frigate guns can usually kill 5x Warrior II before it's tank fails, however it does take a lot of concentration so be ready for it. ECM drones are a different and possibly worse problem. Not only do they prevent you from firing your guns, but they also shut down you scrambler and webifier. All of a sudden the enemy can start manouvering, pull range and then kite you with warp-disruptor range. Death is soon to follow.
3. Energy Neutralization: All frigates have a capacitor disadvantage to cruisers. Energy Neutralizers allow a cruiser to suck the capacitor of a frigate dry. Even for frigates that don't use cap for weapons (autocannons and missiles), no cap means no afterburner, no warp scrambler, no stasis webifier and no armour repper. One way of avoiding neutralizers is to keep range. Unfortunately that is not usually an option (see point 1). Really the only way of dealing with neutralizers is NOS. A small NOS should suck enough cap from a target ship to at least power the scrambler and afterburner. As for repping.... forget it.
4. Stasis Webifiers: If there is one thing that a small ship fears... it is the Stasis Webifier. If the enemy can land a couple of these on you... the whole advantage of being small and fast goes out the window. This is why minmattar recons are so feared. One common 'Fishing' technique for a cruiser is to fit two webifiers. This pretty much locks down a frigate making it easy prey to medium sized guns.
5. Missiles: Missiles do damage with very different calculation than turrets. A small ship can reduce damage from a missile thanks to signature radius and speed, however it is pretty much impossible to drop the damage to zero and still be in a position to do damage in return. Also cruisers have the ability to fit assault missile launchers which are basically frigate based weapons with a higher firing rate. Assault missile Caracals are devastating to a frigate.
The good news for the frigate is that with careful target selection and effective piloting it is possible to successfully kill cruisers. The bad news is that if a frigate can not mitigate the 5 weapons listed above, a frigate will die very very fast.