YE= (Old English) /ji:/ You.
In fact the title is wrong, it should be "God rest you merry", because in Old English YE was the form of subject plural and YOU the form of object plural, later YE disappeared and now we use YOU for both forms in singular and plural. But under the influence of I-ME (and the old singular THOU-THEE) people thought it should be YOU-YE, which is a mistake; it was YE-YOU.

MERRY= (Old fashioned) happy, cheerful.

GOD REST YE MERRY= (Old fashioned) may God rest you merrily, may God make you feel happy and relaxed.
The verb REST here is not a present (in that case it would be "God rests ye"). It is in the subjunctive mood (not in use in modern English) and is used to express a desire. An old surviving expression with the same "present subjunctive tense" is "God save the queen" (= May God protect the queen) or "God bless you".

DISMAY= Disillusion; loss of courage.

LET NOTHING YOU DISMAY= Let nothing dismay you.
In poetry we can often find hyperbatons like this. A hyperbaton is a figure of speech that uses deviation from normal or logical word order to produce an effect. We can find many hyperbatons in this song.

SATAN= The devil; the personification of sin and evil.

GONE ASTRAY= If you go astray you get lost or you take the wrong path.

TIDINGS= (Old English) News.

COMFORT= /kʌmfə*t/

JOY= Happiness.

BLESSED= /blesɪd/ Holy, saint, worthy of worship.

UNTO= (Old fashioned) To.

SHEPHERDS= /ʃepə*dz/ The man who takes care of the sheep (fem. Shepherdess).

BETHLEHEM= /beθlɪhem/ The name of the town where Jesus was born.

O= (Old fashioned) Oh. FEAR NOT= (Old fashioned) Don't be afraid. AFFRIGHT= (Old English) To frighten, to cause you fear, to make you afraid. LET NOTHING YOU AFFRIGHT= (hyperbaton) Let nothing frighten you, let nothing make you afraid. VIRGIN BRIGHT= (hyperbaton) bright Virgin (bright = shining; glorious, splendid). TRUST= To place confidence in someone. MIGHT= (Old fashioned) power. (but in modern English we can still say MIGHTY=Powerful). In colloquial AmE we can also use "mighty" meaning "very", e.g., "that's mighty interesting" = "that's very interesting". LAY= (lie-lay-lain) To be in a horizontal position, resting on a surface.

THE LORD= God. Jesus.

PRAISES= Expressions of admiration (or of adoration to God).

WITHIN= Inside, in.

WITH BROTHERHOOD= With the feeling that we are all brothers and sisters and so we love each other.

EMBRACE= Hold, hug, put your arms around a person.

HOLY= (adjective) Saint, sacred.

TIDE= (Old English) Time: "tide of Christmas" = Christmastide = Christmas time.

DOTH= /dʌθ/ (Old English) Does. In old English, the third person singular in the present added -TH instead of modern -S, so "he loves you" was "he loveth thee" /hi: lʊvɪθ ði:/.

DEFACE= (Old English) Surpass, eclipse, outshine. In modern English this word has a different meaning (= to spoil, to vandalise), but we use the word EFFACE with the meaning of "surpass, eclipse".

DOTH DEFACE= The same as DID PRAY means "prayed", DOTH DEFACE means "defaces" (instead of adding -s for the 3rd person singular we say "does + verb").
The sentence "the holy tide of Christmas all others doth deface" means "the sacred news of Christmas is much superior/better to any other news".