You've probably noticed how the word "honest" shows up all over the place in Othello. By poet and literary critic William Empson's count, there are 52 uses of "honest" and "honesty" throughout the play. Like the word "nothing" in King Lear, "honest" has a wide range of meaning in Othello. At times, it refers to chastity, the question of whether a woman is "honest" or whether she is promiscuous. At other times, the word refers to personal honesty, whether or not a person is telling the truth. It can also refer to whether or not a person is a good and loving friend.
These meanings come together in some ironic ways throughout the play. The clearest example of this is how Iago uses personal dishonesty (lies and deceit) to convince Othello that his wife is sexually dishonest (cheating on her husband), all while pretending to be looking out for the best interests of his so-called friend. Check out how Iago plays the martyr when Othello warns him that he, Iago, better not be lying about Desdemona:
O wretched fool.
That livest to make thine honesty a vice!
O monstrous world! Take note, take note, O world,
To be direct and honest is not safe.
I thank you for this profit; and from hence
I'll love no friend, sith love breeds such offence.
Nay, stay: thou shouldst be honest.
I should be wise, for honesty's a fool
And loses that it works for.
By the world,
I think my wife be honest and think she is not; (3.3.52)