Damage-Per-Second, also referred as DPS, means the amount of Damage you can deal every second. Damage-Per-Minute, also referred as DPM, means the amount of Damage you can deal every minute. DPS and DPM work the same, except that the time-frame is different (1 second or 60 seconds).
Small cannons deal more DPS than big cannons. Thus, if you want to deal consistent (constant) damage, small cannons are better. However, if you want to have Spike Power, big cannons are better. The middle type cannons (12lb, 16lb and 18lb) have a nice mixture of both.
To increase your DPS, you can use Fittings, Skills and Gunpowders that either give Damage or Reload. Also, Accuracy does increase your DPS, because it increases your chance to hit, thereby doing more Damage.
For more information, look at the table below*.
This information is provided by the update that PotBS Wikia is undergoing, by Captain Vuur.