These lines are spoken by King Duncan in act 1, scene 4 and refer to the duplicity of the former Thane of Cawdor, who has just been executed for treason. Malcolm praises the Thane of Cawdor, saying that the man who did not live well was at least capable of dying well, since he showed both genuine repentance and courage when facing death. Duncan reproves and contradicts his son here, saying that appearances are deceptive. There is no way to know what a man is really thinking and feeling simply by looking at his face.

The irony is that the new Thane of Cawdor, who is about to enter, is to prove even more deceptive and deadly. Later in the same scene, Macbeth is already thinking about the need to conceal his true nature:

Stars, hide your fires;
Let not light see my black and deep desires:
The eye wink at the hand; yet let that be,
Which the eye fears, when it is done, to see.

When, in the following scene, Lady Macbeth is encouraging her husband to kill the king, she makes the opposite point to the one that occurs to Duncan here. In her opinion, it is all too easy to see what Macbeth is planning from looking at his face:

Your face, my thane, is as a book where men
May read strange matters. To beguile the time,
Look like the time; bear welcome in your eye,
Your hand, your tongue: look like the innocent flower,
But be the serpent under't.

There is an ironic contrast here between the old king, who should be battle-hardened and cynical but remains innocent, and the lady, who is supposed by conventional standards to be innocent, and is clearly able to look so, but whose appearance is deceptive. She may be able to read Macbeth's face like a book, but Duncan has already declared himself incapable of doing so.

When King Duncan speaks this line from act 1, scene 4 of Shakespeare's Macbeth, he has just been informed that the Thane of Cawdor has been executed. Malcolm tells the king that the men in charge of the execution have not returned from the task yet, but that he (Malcolm) talked to someone who saw Cawdor's execution. This messenger said that Cawdor admitted his crimes, deeply repented, and asked King Duncan's forgiveness. Then the messenger reported to Duncan that Cawdor died very well. At the end of Malcolm's speech, King Duncan responds:

There's no art to find the mind's construction in the face. He was a gentlemen on whom I built an absolute trust.

An art in this context means a creative skill. We use this word in several contexts. For example, there is an art and a science to teaching and practicing medicine. This word art also refers to magic arts, like mind reading or fortune-telling. What King Duncan means by this is that one cannot read another's mind by looking at his face. King Duncan had placed absolute trust in the Thane of Cawdor, therefore his betrayal took him completely by surprise. 

The description of the Thane of Cawdor's betrayal comes in act 1, scene 2 when Ross tells the king that the Norwegian flag is flying in Fife, and he saw the Norwegian king leading an immense army and being assisted by the Thane of Cawdor.

The irony of King Duncan's statement about not being able to read someone's mind based on their face is that he will be betrayed to an even greater degree by Macbeth, whom he is also unable to read and in whom he placed absolute trust.

To understand the meaning of this line, it is important to consider its context. This line comes from Act I, Scene IV, and is spoken by King Duncan. Here, Duncan is speaking with his noblemen, including Macbeth, about the Thane of Cawdor and his treason. Specifically, Duncan is saying that he had no idea that Cawdor would turn out to be a traitor. After all, it is impossible ("there's no art") to know what a person is thinking ("the mind's construction") simply by looking at the face, because a person can hide his true intentions and feelings.

Looking deeper, this line is just as applicable to Macbeth as it is to Cawdor. In the previous scene, for example, Macbeth listened intently to the witches, who prophesied that he would become the next king of Scotland. This prompted to Macbeth to suddenly start thinking about murdering King Duncan. But, just like Cawdor, Macbeth hides these murderous thoughts. In this scene, Macbeth acts completely loyal and trustworthy to Duncan.

So, for the second time, Duncan is unable to read the "construction" of a person's mind. This time, it is Macbeth's mind, and Duncan is completely clueless about the impending murder.

This is a line that is spoken in Act I, Scene 4 by King Duncan.  He is talking to Malcolm after Malcolm has reported to him that the Thane of Cawdor has been executed.  What Duncan means by this is that there is no way to tell what a man is like -- what his character is -- by looking at his face.

When Duncan says "there is no art" he means "there is no way."  When he refers to a "man's construction" he means the man's character.

What he is saying is that Cawdor basically fooled him -- he thought Cawdor was loyal but he turned out to be a traitor.

Explain the quote "There's no art to find the mind's construction in the face."  How does it relate to Lady Macbeth in Macbeth?

In Act l, scene lV, King Duncan speaks these lines about the Thane of Cawdor, who has betrayed him. He has just discovered that Cawdor was executed after he confessed his treason and begged the king's forgiveness.

What Duncan means is that no one can determine what another is thinking by looking at his or her appearance or the expression on such a person's face. It is impossible, therefore, to judge whether the person bears you any ill will or not by reading his or her face. He acknowledges that the Thane of Cawdor deceived and misled him. He trusted the thane and believed that he was loyal. This fact is further supported by his earlier statement in Act l, scene ll, when he ordered that Cawdor be killed and Macbeth be given his title:

No more that thane of Cawdor shall deceive
Our bosom interest.

Lady Macbeth, from the outset, is intent on misleading others. She, for example, commands her husband to "appear the innocent flower but be the serpent underneath." She tells him that he must put on a face of conviviality and friendship when he meets Duncan and his entourage when he is, in fact, plotting his malice. Indeed, when she welcomes the king to their castle at Inverness in Act l, scene Vl, she puts on a show and warmly welcomes him:

All our service
In every point twice done and then done double
Were poor and single business to contend
Against those honours deep and broad wherewith
Your majesty loads our house.

Her exaggerated welcome hides her true intentions. She has already decided that Duncan should be murdered and that his visit will give her and her husband the ideal opportunity to enact their perfidy. Prior to this, she called on the powers of darkness to hide her malicious intent and empower her to commit evil. 

It is obvious that Lady Macbeth is sly and ruthless. She is intent on implicating others by creating the illusion that they are responsible for what she and Macbeth are about to do. She informs her husband that:

. . . what not put upon
His spongy officers, who shall bear the guilt
Of our great quell?

She intends getting Duncan's guards so drunk that they will be incapable of defending him during the assassination. They will not remember anything and will be easy to blame for a crime that she and her husband are about to commit. She and Macbeth will, after "learning about" Duncan's murder, put up such a show of grief that others will be convinced of their loyalty and love for their now-deceased liege and will, therefore, not suspect them of any wrongdoing.

In addition, she has convinced Macbeth that their task will not fail. He eventually succumbs to her will and agrees about putting on an act. He states, at the end of Act l:

False face must hide what the false heart doth know.

Lady Macbeth's conniving and plotting tie in well with the theme of appearance and reality. Her sole purpose is to illegitimately gain power. She will deceive and mislead to ensure that she and her husband achieve their goal. It is ironic that she is the first to eventually succumb to remorse. She is overwhelmed by guilt and becomes so overwrought that she commits suicide. 

Explain the quote "There's no art to find the mind's construction in the face."  How does it relate to Lady Macbeth in Macbeth?

The quote you reference is originally spoken by King Duncan in Act 1, Scene 4 of Shakespeare's Macbeth. In context, the quote references how the former Thane of Cawdor fooled Duncan and betrayed him. Basically, in this quote Duncan is acknowledging how difficult it is to know what's going on in someone else's mind based on their outward appearance. In other words, it's easy for someone to hide his or her real feelings in order to betray someone.

This quote could certainly apply to Lady Macbeth on multiple occasions, as she proves adept at hiding her true feelings for most of the play (until, of course, she's goes crazy and dies). The most obvious example of this is Act 2, Scene 3, at which point King Duncan's murder is discovered. At this point, we know Lady Macbeth had a role to play in the murder. That said, you wouldn't guess it from the way she pretends to be surprised, and then emotionally/physically weakened by the grisly "revelation." Indeed, Lady Macbeth carefully hides her guilty conscience by pretending she is too "weaK" to deal with the news of Duncan's murder. As such, at this point in the play Lady Macbeth's outward appearance certainly masks her "mind's construction."