Mary Betrothed to Joseph

What is meant by Mary having been “betrothed to Joseph” (Matthew 1:18)?
She was legally bound to marry Joseph, and he was legally bound to marry her. An official divorce was required to break a betrothal, which was more binding - notice Joseph is already being called Mary's “husband” (Matthew 1:19 below) - than today's “engagement.” For details on how a betrothal worked, see Bride and Bridegroom.

What is meant by “before they came together” (Matthew 1:18)?
Before they married and consummated their marriage by sexual intercourse.

What is meant by Mary being “found to be with child by the Holy Spirit” (Matthew 1:18)?
God the Holy Spirit made Mary be pregnant without a man having sex with her.

How can the Creator God do that?
 

How did Joseph learn that Mary was pregnant?
Since he intended to send her away “secretly” (Matthew 1:19), her pregnancy wasn't public knowledge; Mary herself most likely told Joseph that she was pregnant.

How must Joseph have felt upon hearing the news?
How would you have felt if a young virgin girl whom you loved and was looking forward to marrying and consummating the marriage with told you that she is already pregnant? Joseph most likely was at least devastated and probably also upset at what he thought to be her adultery.

Didn't she tell him that she became pregnant by the power of the Holy Spirit, and not by cheating on him with another man?
She probably did, but would you have believed her story if you had been in Joseph's shoes?

What is meant by Joseph being “righteous” (Matthew 1:19)?
The original Greek word δικαιος (dikaios) also means "just" or "upright." Joseph wasn't someone who could simply turn a blind eye to what he thought to be Mary's sin of adultery. But neither did he want to see her publicly disgraced, let alone executed as per Mosaic law: “If a man is found lying with a married woman, then both of them shall die - the man that lay with the woman, and the woman; so you shall purge the evil from Israel” (Deuteronomy 22:22; see Woman caught in adultery).

Why not?
He most likely loved her.

So what was he planning to do?
Privately write her a certificate of divorce and “send her away secretly” (Matthew 1:19) as also allowed by Mosaic law: “When a man takes a wife and marries her, and it happens that she finds no favor in his eyes because he has found some uncleanness in her, and he writes her a certificate of divorce, puts it in her hand, and sends her out of his house, ...” (Deuteronomy 24:1).