Equity is how a checkout line (queue) works: people line up and are served in order. This is also know as "first-come, first-serve." Notice that we don't randomly select among the customers waiting to be served (equality) nor do we order the customers by the number of items in their cart (relative need).
Equality is how a one-ticket-per-person raffle works. Assuming the winning ticket is chosen at random, the probability of winning is the same for each ticket-holder. We don't award the prize to the first person who buys a ticket (equity), nor do we award the prize to whomever wants or needs it the most (relative need).
Relative need is how an emergency room works. If Alice is waiting to have a splinter removed and Bob needs immediate surgery, the relative needs of Bob take priority over the needs of Alice and Bob is served first. While in some cases, we'd process patients in the order in which they arrived (equity), it seems rather arbitrary to select patients at random (equality).
A simple example is affirmative action. Both sides of the debate tend to complain about "fairness," yet both refer to different policies. One side demands equality or equity, while the other wants a policy of relative need. Of course, such realizations do not solve the problems involved, but they do clarify the locus of dispute.
How do we decide which policy of "fairness" to apply in a given situation?