Psalm 42 is the beginning of book two in the overall structure of the book of Psalms (see the Bible Project video below for the overall structure of the book). It is also the beginning of what is known as the Elohistic Psalter... that name just flows off the tongue doesn't it! Basically, in these Psalms the word God (Elohim) is used most frequently instead of the personal name for God, YAHWEH, usually translated The LORD.
The beginning of Psalm 42 notes that it is "for the director of music. A maskil of the Sons of Korah." First, a maskil is a song that is meant to teach. This is common even today. Many of our songs contain lyrics that are more easily remembered when put to song and also teach important spiritual truths.
The Sons of Korah were temple keepers and singers during the reign of Solomon and David according to 1 Chronicles (1 Chron. 6:31-37; 9:19). They were one of the five major Levitical families who were charged with the priesthood office and performing the duties required of worship in the temple. The Korahites performed some of the more menial tasks required at the temple, this due to an incident in Numbers 16 in which Korah inappropriately challenged the authority of Moses.
Due to the implication of the loss of temple worship in v. 1-4 this psalm is most likely written during the Babylonian exile. In 586 BC Babylon attacked the southern kingdom of Judah and overthrew it, destroying the temple and taking many Israelites to Babylon in exile. This was an incredibly dark time in the history of Israel. See the Bible Project page below for a more thorough description of the exile period in Israel's history. It could be a number of other incidents where segments of Israelites were taken from their home by an invading force, but the exile seems to make the most sense.