A song of ascents, by David. 

Psalms 120-134 are called “the psalms of ascents.” A better name would be “the songs of the pilgrim caravans.” These pilgrimage songs were sung when Israel traveled upwards to Jerusalem (usually in caravans) during the Jewish feasts of Passover, Pentecost, and Tabernacles. Four of these songs were attributed to David. Solomon wrote one of these songs. The other ten songs are anonymous. The author of Psalm 133 was written by David.

133:1 Look! How good and how pleasant it is when brothers live together! 

The sons of David were continually at war with each other. Each of them wanted to become king. They committed incest and murdered each other. David asked his sons to live in peace with each other.

133:2 It is like fine oil poured on the head which flows down the beard— Aaron’s beard, and then flows down his garments. 

Aaron was anointed with oil when he became the first High Priest of Israel. The oil flowed down his head, down his beard, down his shoulders, and down his breastplate. The breastplate of the High Priest contained the names of the 12 tribes of Israel. Aaron and the twelve tribes were anointed to become the priests to all of the nations. Israel will not fulfill this priestly commission until the Great Tribulation. 

133:3 It is like the dew of Hermon, which flows down upon the hills of Zion. Indeed that is where the Lord has decreed a blessing will be available—eternal life. 

When the dew runs down the Mountains of Hermon, then all of Israel is blessed. If David’s sons would get along with each other, then the Spirit of God would pour down upon the nation of Israel just as the dew poured down Mount Hermon.