In School I learned the definitions and proved theorems with the directed sets introduced by Moore and Smith. I did not then understand why these notions were supposed to be superior to the earlier conventional sequences of points in a topological space. Kelly introduces these notions with enough background to motivate them. I expound here.

directs D iff ≧ is a relation on D and:

In other words ≧ is transitive and reflexive and directed.

A directed set is (D, ≧) where ≧ directs D.
A net in a topological space is a function S from a directed set to the space.
A net converges to a point z in the space iff ∀v (v is a neighborhood of z) → ∃p∊D ∀q∊D (q≧p → S(q)∊v).

The integers ordered by the ordinary ≧ satisfy these conditions.

Nets are better than sequences because:

In short the rules for nets are simpler, even though some nets are necessarily more complicated than the integers.