Damage over time is the name given to several conditions which do damage over several turns. Each one functions similarly, but they deal different damage types, and so are given different names.
Each "Damage over time" condition is given a number, such as in "Diseased 5". At the start of the turn, the champion takes damage equal to the number (how many turns are left) and the number decreases since a turn has passed. So Diseased 5 becomes Diseased 4, then Diseased 3, and so on. Thus, if it is still alive when it is Diseased 1, at the start of the next turn it takes 1 damage and the Diseased condition is removed.
Assuming the champion survives and normal damage, Duration 3 = 3+2+1=6 damage. Duration 4 = 4+3+2+1=10 damage. Duration 5=5+4+3+2+1=13 damage.
A damage over time effects is frequently referred to as a "DoT"
Some damage over time effects also cause debuffs.
Poison causes -1 DEF and -1 DMG.
Scour causes -1 DEF.
Diseased causes the condition Inhibited.