Those mouth speaketh vanity - Vain things; things not real and true; falsehood; lies. See the notes at Psalm 24:4. The idea is, that what they said had no foundation in truth - no reality. Truth is solid and reliable; falsehood is unreliable and vain.
And their right hand is a right hand of falsehood - The meaning here seems to be that even under the solemnities of an oath, when they lifted up their hands to swear, when they solemnly appealed to God, there was no reliance to be placed on what they affirmed or promised. Oaths were taken by lifting up the right hand as toward God. See Genesis 14:22; Exodus 6:8 (Margin, and Hebrew); Deuteronomy 32:40.

Whose mouth speaketh vanity, and their right hand [is] a right hand (g) of falsehood.
(g) For though they shake hands, they do not keep their promises.

Whose mouth speaketh vanity,.... Vain words, lies, flatteries, and deceit, Psalm 12:2; when they speak loftily of themselves, and contemptuously of others; when they deliver out threatenings against some, and make fair promises to others; it is all vanity, and comes to nothing;
and their right hand is a right hand of falsehood; their strength and power to perform what they boast of, threaten, or promise, is fallacious, is mere weakness, and cannot effect anything; or their treaties, contracts, and covenants, they enter into and sign with their right hand, are not kept by them; they act the treacherous and deceitful part. The Latin interpreter of the Arabic version renders it, "their oath is an oath of iniquity"; and Ben Balaam in Aben Ezra, and R. Adnim in Ben Melech, say the word so signifies in the Arabic language; and Schultens (m) has observed the same: but the word in that language signifies the right hand as well as an oath, and need not be restrained to that; it is better to take it in the large sense, as Cocceius (n) does; whether they lifted up the hand to pray, or to swear; or gave it to covenant with, to make contracts and agreements; or stretched it out to work with; it was a right hand of falsehood.
(m) Observat. Philolog. p. 195. (n) Lexicon, col. 312.

Vanity - Vain brags and threatenings which shall come to nothing. Falsehood - Deceiving themselves, by being unable to do what they designed; and others, by not giving them that help which they promised.