UP is a most basic principle in moral reasoning. It says:

If one makes  a moral judgment about a particular case, then one must make the same moral judgment is other similar cases, unless there is a relevant  difference , a difference which is relevant to the moral issues involved in the cases at hand.

• "It would be wrong for me to cheat; UP; hence it would be wrong for you to cheat" is invalid, because the argument doesn't show that there are no relevant moral differences between our two cases.
• "It would be wrong for me to cheat, but not for you because there's a significant difference between us: I'm Chinese and you're French" is unacceptable because the difference, though significant, is not shown to be morally relevant.
• "I shouldn't pay taxes but you should because I'm a morally good person and you're not" is unacceptable because, although there's a moral difference, there's no evidence that it is relevant.
• UP creates a strong presumption against discrimination on the basis of sex or race, since in most cases it's hard to see what relevant moral differences exist among sexes and races.

Why accept UP?
This is a difficult question.  A possible answer, due to Hare, is roughly as follows.  Terms such as "good", "bad", "right", "wrong" are descriptive, just like "red" is.  But the meaning of a descriptive term doesn't change when applied in different situations, and therefore the criteria of application do not change either.  For example, "box A is red, and box B is exactly like box A in all relevant color respects, but it isn't red" strikes us as absurd.
Similarly, "Act A is right (wrong), and act B is exactly like A in all relevant moral respects, but is is wrong (right)" should strike us as absurd.
Note that UP is not in itself a moral principle because by itself it tells you nothing about what to do or not to do (consider the case of the consistent Nazi or the consistent Hutu genocidal murderer); rather, it provides a consistency requirement for moral agents.

The Golden Rule (GR)
GR says that

we must act (on others) as we would be willing to have others act (on us), or equivalently by contraposition, that we must not act (on others) in ways that we are unwilling to have others to act (on us).

It's a very old rule, first codified in ancient Mesopotamia, and later on in the judeo-christian-muslim tradition.  In fact many religions adopt it. Many take GR to provide a foundation for morality. However, it's hard to see how it can fulfill such role. GR can be understood in different ways.

A. GR can be taken to say that I should act towards you as I would be prepared to accept you acted towards me if our places were exchanged. The problem here is given by cases like the consistent Nazi: there are many people who are prepared to have awful things done onto themselves, but this gives them no justification to do them onto me.

B. GR can be taken to say that I should act towards you as I would wish you acted towards me if our places were exchanged. This formulation falls prey to Whatley criticism, who argued that GR, thus understood, involves is sometimes absurd, sometimes wrong, and sometimes impossible.

• Absurd: The landlord would be glad to have his rent remitted if he were the renter. Hence, by GR, he should excuse his renter from paying, which is absurd.
• Wrong: a jailer, if he were the prisoner would want the jailer to let him escape. Hence, the jailer should let the prisoner go, which is wrong.
• Impossible: The judge between A and B knows that each of them want him to judge in their favor. Hence, he should judge in the favor of both, which is impossible.

C. GR can be taken to say that I should act towards you as I would fairly expect you acted towards me if our places were exchanged. This takes care of the prevoious objection, but the notion of "fair expectation" presupposes morality.

GR, like UP, may provide a test for moral consistency by aking us consider whether we are blinded by self-interest.