What does "To thine own self be true" mean?

Before Laertes leaves, his father, Polonius, offers him quite a bit of advice. He counsels his son to keep his thoughts to himself, to be friendly but not vulgar, to be a loyal friend, to avoid fighting but to fight hard if it cannot be avoided, to listen a lot and speak little, accept the criticism of others with grace but do not criticize others, dress nicely but not too richly, to not borrow money because it results in bad spending habits, and to not loan money because it results in lost friends. Finally, Polonius says, "This above all: to thine own self be true" (1.3.84). For if one is honest and true to oneself,

it must follow, as the night the day,Thou canst not then be false to any man. (1.3.85–86)

If one is true to oneself, then one will be truthful with other people too. If Laertes follows his own conscience rather than the example set by others or influences from society at large, he will make decisions that will make his father proud. If, on the other hand, he chooses to follow another's lead rather than his own, he will misstep.

A very common allusion to Shakespeare's Hamlet is found in the common truism: "This above all: to thine own self be true." Ironically, this little bit of wisdom is actually spoken by Polonius, the fool of the play. Nonetheless, these words have something important to impart. These words are a departure from the silly explanation of proper etiquette given in the first half of Polonius's long speech.
This above all: to thine own self be true, / And it must follow, as the night the day, / Thou canst not then be false to any man (Act I, Scene 3, lines 77-79).
Put simply, if you are true to yourself and who you really are, just as the sun always rises each morning, then there is no way you will be a liar to anyone you meet. You will always project truth because it will emanate from your very being.
A bit of irony here is that immediately after Polonius instructs his son to be true to himself, he instructs his daughter otherwise. Ophelia spends her first words of this play trying to show her dad how Hamlet is quite an honorable potential husband by saying that "he hath, my lord, of late made many tenders / Of his affection to me. . . He hath importuned me with love / In honorable fashion." Polonius immediately causes Ophelia to doubt herself, "Do you believe his tenders, as you call them?" Ophelia reneges and promises to "obey" her dad in not receiving Hamlet's advances. So much for being true to oneself.