How is the word "honest" used in Shakespeare's Othello?

Well, for starters, Shakespeare used the word "honest" or some form of it a lot in Othello: over fifty times! So it's definitely worth our consideration.

First, the characters talk about who does or doesn't have an "honest face" or an "honest hand." In Act 2, Scene 3, Montano says it would be an "honest action" to tell Othello about how his second-in-command is an alcoholic. And there's a lot of talk about Desdemona and other women being "honest" or not. You get the idea that Shakespeare's use of the word "honest" in this sense is a stand-in for all kinds of virtues: like openness, faithfulness, and chastity. This makes sense if you take a look at the entry for "honest" in the Oxford English Dictionary and note that, in addition to "truthful," it also used to mean "noble" and "respectable" as well as "virtuous" and even "chaste."

Second, characters in this play sometimes call each other an "honest knave" or an "honest fool." You'll also find the phrase "honest plainness" in the play. In phrases like this, Shakespeare is using "honest" to mean "sincere" or "genuine." Again, the Oxford English Dictionary reveals that these, too, are valid definitions of the word "honest."

Third, and most importantly, many of the characters including the villain Iago himself constantly refer to him as "Honest Iago." In this case, Shakespeare is using the word ironically. It's funny; it's meant to entertain us: by continually referring to this villain as an honest guy, the characters are calling attention to Iago's deeply twisted, manipulative personality. And yet in some situations, Iago does often tell the truth--which only makes him more dangerous, since Othello doesn't know when he can trust Iago and when he can't.

The issue of Iago and his honesty becomes especially important in Act 3, Scene 3, when Iago is trying to get Othello to believe that his wife is unfaithful--and Iago calls attention to his own "honesty." He calls himself a fool for always telling the truth, which is ironic considering how he's currently lying about Othello's wife:

O wretched fool

That lov’st to make thine honesty a vice!

O monstrous world! Take note, take note, O world,

To be direct and honest is not safe.

Iago is seriously taking advantage of his own reputation as someone "honest" here: an incredibly dishonest thing to do.

What are some important quotes regarding Iago's "honesty" in Othello?

You are quite right to put "honesty" in quotation marks, for any mention of Iago as being honest must be ironic.   However, the irony is mostly dramatic.  All of the major characters believe that Iago is honest, so when the term is used, the audience knows that each of these characters has misjudged Iago.

One of the first references to honesty is made by Iago himself, in the last few lines of Act 1.  He characterizes Othello as one who

is of a free and open nature

That thinks men honest that but seem to be so

A perceptive judge of human nature, Iago understands that Othello is truly honest, and because he is so, he sees others as he is himself.  Thus, Iago knows that Othello will trust him.

Later, Cassio calls Iago "honest" after Iago gives Cassio advice that will result in Othello's ordering Cassio's death:

Good night, honest Iago.

But the term reaches its highest irony when Othello calls Iago honest:

And, for I know thou'rt full of love and honesty

And weigh'st thy words before thou giv'st them breath.

And later, Othello declares,

This fellow's of exceeding honesty,

And knows all qualities, with a learned spirit

Of human dealings. (Act 3, scene 3)

These references to Iago's honesty, while initially are somewhat humorous because the audience knows how grossly Othello has misjudged Iago and his intentions, are the key to understanding why Othello falls for Iago's machinations and turns against his most beloved wife Desdemona.  It is Othello's faith in those around him, his tendency to see others as he himself is, his trust in the men who have served with him in battle that leads to Othello's downfall.  In other words, much of what causes Othello to turn against Desdemona is his inability to see the darker nature of Iago.

In Othello, how does Iago deceive others and why is he called "honest Iago"?

Outwardly, Iago seems honest but he is actually a very petty and small person. His main ploy is playing on other people's insecurities. His interactions with Othello are a prime example. Othello is an honest and virtuous man but has the flaw of jealousy. Iago's tone and persona is of a very honest and respectable man. In this way he is able to lure Othello into his trap. The first half of the play is very much Iago's. Most of the action is seen through his point-of-view. It is important to note that the audience sees a side of Iago that the other characters don't. Namely that he is scheming their destruction. In this way, his "honest" persona is clearly ironic. It is a ploy which he uses to his advantage. At the end of the play he reveals that he destroys because he destroys because he can't stand that Cassio "will have beauty in his life." Ultimately, his motives are banal. The key to Iago's success is that people trust him because he tells them what they want to believe because he is perceptive to their weaknesses.

Is Iago "honest" in Shakespeare's Othello? Explain.

The question speaks to Iago's morality. Is his 'honesty' a measure of his moral turpitude or his moral supremacy? Yes, he is honest to the audience throughout the play and speaks of his devious nature and makes a declaration of intent, without apologising for what he does and what he intends doing. His sole desire is to destroy those by whom he feels wronged or threatened, by whatever means possible. His devious machinations are purposeful and a means to an end: He follows Othello to 'serve my turn upon him.' Iago, for all his honesty, does not care about how his actions would affect innocent victims, they become mere residual damage in his journey of destruction.

In reality, the fact that Iago is honest in his declarations to Roderigo further emphasises his evil, for he does this only to gain Roderigo's support and the gullible fool realises too late that he had also been manipulated and had been a mere tool in Iago's devious plan.

Added to this, we the audience, are impressed by the fact that Iago is honest to himself and does not make excuses for who he is. But this does not make him a better person, for the audience has no influence on his actions and therefore his honesty, in this regard, has no merit whatsoever. Shakespeare cleverly uses Iago's honesty with the audience as a device to add to the dramatic irony and therefore the tension. The more Iago confesses to us, the greater the impact. We wait in expectation for his next act of evil and its resultant effect on the unfortunate characters toward whom his perfidy is directed.

Furthermore, Iago at times, functions as a narrator and his glee in confessing his evil makes him a more pernicious individual, for he takes pride in the fact that he can manipulate others so easily and make them believe his every word. He comes across as arrogant and psychotic, not admirable.

With the exception of Roderigo, Iago does not confide in anyone else about his intentions, for he fears that they could turn against him. This makes him a coward. In effect, he does not admit his cowardice for he deems it a weakness, whereas his exploitation and manipulation is seen as a strength, something to brag about to those from whom he can expect no intervention. His 'honesty' is therefore selective. When he does discover a threat to his machinations, when Roderigo for example rebels against him, he kills him and blames others for his murder.

The fact that others trust him and believe in his honesty, is a source of great pride and satisfaction to Iago and this is where the measure of his character is grandly displayed. His malice becomes more evident for he uses this belief in him and his goodness to drag them further into his web of lies and devastation. He carries this through right to the end and never admits to his wrongdoing. When he is finally caught out, he like the coward he is, declares:

Demand me nothing: what you know, you know:
From this time forth I never will speak word.

So, no, Iago is not honest. The superficial and meaningless confessions he makes when addressing the audience and Roderigo, is not honesty at all. It merely displays the depth of his evil, for he realises that such 'honesty' does not present him with any threat. He can freely execute his malicious and vindictive plot.