NOTE: Assisting does not allow a character to use targeted powers on herself, nor an enemy that has too much Stealth for her to see.
Attacking Through Ally
If a character attempts to use a harmful power while targeting an ally, or activates a helpful power on a hostile target, she will automatically use that power on her target's target instead, if it has one, if this secondary target is legal for her power, and if the secondary target is within range. The game does not move the targeting reticle to the new target.
This feature is often used with ranged attacks by Blasters or Corruptors in PvE teams with a Tanker or Brute, to guarantee those blasts strike the same enemy that that tank is taunting. Assisting can also be used in PvP to ensure a coordinated attack on a single enemy. These techniques are also known as "attacking through" someone.
Assisting can also be used to automatically use a buff or healing power on whichever teammate a certain enemy is focused on, without needing to guess who that is or waiting to see who gets hurt.
Switching to Ally's Target
Additionally, utilizing the /assist slash command allows a character to switch targets from an ally to whomever the ally is targeting, including ally or hostile targets.
Not Related to Targeting
Context is important; assist can also mean simply to help another player with a mission. This is the most generic use of the term.