Damage per Second
DPS is an abbreviation for "Damage per Second". It's an average of much damage to targets HP that's done/is going to be done/might be done per second. As an example is if you did 6000 damage in 60 seconds the DPS would be 100HP/s.
This example above is of actually dealt DPS and as such the DPS is calculated from what actually happened. That is not very useful because you're either dead or happy already. What we are interested in is theoretical DPS. You can of course calculate this very close to the dealt DPS value but we usually have to leave quite a few variables out like for example speed and direction. People in Eve nearly always simplify it even more, either because target isn't known or a more complicated average is... well, more complicated. So we often also remove resistance, sig radius, reload time, tracking speed, hit chance etc.
DPS in Eve-chats are almost always calculated like below. It's excluding most of the variables and it's great for creating a feeling of the ships capabilities. But it does suffer some problems like you can have a high conventional DPS value but your hit chance might be close to zero. Which would result in an actually dealt DPS of close to zero. Which is the problem when people use software like EFT without thinking.
(Turret Damage modifier*Ammo Damage*Number of Turrets)/Rate of Fire
(Ammo Damage*Bonus from Ballistic Control Units)/(Rate of Fire+Flight time) where flight time is excluded sometimes.
The DPS for drones is a bit more work as it doesn't factor in skills and stuff on the ship that affects damage.
The easy way
You can use EFT to do the above calculations for you. EFT Also offer this DPS with regards to range. Or you can use EveHQ that also offers some more intricate data.
Complicated side note on conventional DPS:
When you fire a turret you fire in the beginning of the cycle so if RoF is 3 sec you will actually fire 2 times in close to the first 3 seconds (depending on lag). So the DPS starts higher and converges to the values in "conventional DPS" above as time moves towards infinity. This can be good to know as PvP fights usually doesn't last very long. Here's an example:
Below is the DPS for the example above calculated after each shot is fired. After the first 4,05sec the turret will have fired 2 times. After 8,1 sec it will have fired 3 times.
More basic info:
Check out Evelopedia for more basic info (that for the moment seem to be totally wrong).
DPS is depending on damage and damage depends on a lot of variables. What variables do you know? Are you sitting still outside a station hitting specific targets? Then you know range, approx ang, sig rad etc. Damage and hit chance for turrets is calculated with this formula. Check out this link discussing calculating the DPS for turrets with regards to hit chance. The damage for missiles is also done with a formula but always have 100% hit chance if it catches you. Drones? No idea sorry.
Thought for the day:
Ask yourself why you are calculating DPS. Are you going to hit certain T3s undocking from a station exactly 30 km away while standing still? Then you might want to calculate a target-specific DPS with sig radius and range etc. Are you going to engage in PVP against unknown targets? Then you might want to go for a more simplified DPS-calculation. Are you bragging about your DPS or comparing tank-capability? Then go for the conventional calculation. Know this though: the more variables you factor in the less chance is it that you have a surprised look on your space-frozen body. Knowledge is power.