What do IVs actually do?

IVs cause a pokemon to have slightly better or worse overall stats, depending on how good the IVs are. Pokemon have base attack, defense, and stamina that are determined by their type. IVs add to these base values. Attack is a factor in how hard a pokemon hits, defense in damage taken, and stamina determines how many hit points it has.

A 1000 CP snorlax with 50% IVs would have to be higher level than a 1000 CP snorlax with 100% IVs. They would be around the same at fighting. But if they were the same level instead of the same CP, the 100% IV one would be better.

IVs are added to the base stats that you see listed on this site.

It simply means a mon will have higher stats compared to another mon of the same species when they are at the same level.

In practice, this means that a mon with higher IVs can reach higher max CP and can reach a high CP at a low level.

Level 35.5
Vaporeon 14/15/14 (95,6%)... 2941 CP
Vaporeon 9/15/13 (82,2%)... 2906 CP

Gyarados 14/15/10 (86,7%)... 3026 CP
Gyarados (red) 14/5/10 (64,4%)... 2884 CP
Yes, the red would need to be three complete levels (6 ups) more to reach the other one.

As others said, IVs increase the stats. These stats not only figure into the CP, but also into the damage dealt and received during battle. Read the article "Damage Mechancis" on this page for details.

In my experience, it really only matters for the mons that you max out above L30. If you like a mon enough to power him up from 35 to 35.5, you are using a lot of dust and candies for a very small bump in CP. It is at this point where I always wish I had a higher IV mon.

For attackers:
IVs dont really matter much but having the best move set possible does (highest dps)! Try to do some simulaton battles yourself here: https://pokeassistant.com/battlesimulator

For defenders:
IVs do matter because the higher the IVs (especially the attack IV!), the higher the CP goes and in the current meta, I want to be placed in gyms as high as possible. Doesn't really matter if it's a 'good' defender or not because if someone wants to take down a gym regardless of it's LVL, he will get it down no matter what pokemon it holds - A LVL10 Blissey gym could make a difference tho :p.

If they both have 1000 CP, than yes they are roughly equals.
As others pointed, they had to be in different levels for this to happen.