The Authors of the Psalms

Although the Psalms are often attributed to David, slightly less than half (72 of 150) are really identified as originating with him. A number of them are anonymous but others are attributed to several bands of temple musicians. According to 1 Chronicles 6: 31-47, and 1 Chronicles 25: 1-7 David put some men and their sons in charge of temple music. This included Asaph, Jeduthun and Heman. Heman was grandson of Samuel (1 Chronicles 6: 33.)

David is listed in the heading of 73 psalms. All but psalms four psalms (1, 2, 10 and 33) of the 41 psalms in Book I are described as "of David." (Also, when the Christian disciples pray in Acts 4: 25 they quote from Psalm 2, a messianice psalm, and cite the author as David, although "David said..." could be a euphemism for "The psalmists said...")

In Book II of the psalms, David is described as author in 18 psalms: 51—65 and  68—70.

In Book V, David is described as author of 15 psalms 108—110, 122, 124, 131, 133, 138—145.

Three remaining psalms,  86, 101, and 103, also describe him as author.

The Sons of Korah

In Numbers 16, a man named Korah instigates a rebellion against Moses. Eventually Korah and his rebellious party are destroyed by God but Korah's sons are spared (Numbers 26: 9-11) and fathered a line of descendants called the Korahites.  Since Korah was a grandson of Kohath (Numbers 16: 1), the Korahites were part of a larger band of Kohathites. 

Korah's descendants are assigned as a group in charge of temple music and the first eight psalms of Book II (that is, Psalms 42-49) have a heading that attributes them to the Sons of Korah.  There are five more psalms in Book III (Psalms 84-88) attributed to the Sons of  Korah. It is likely that the Sons of Korah was a guild that lasted for numerous generations.

Asaph

Asaph is mentioned in 1 Chronicles 6 (link above) as one in charge of music and twelve psalms are arttributed to him: 50, 73—83. (These last eleven psalms start Book III of the Psalms.)

Other authors

Heman, a grandson of Samuel, is listed in the 1 Chronicles 6 passage, as one of David's temple musicians. He is also described, along with Ethan the Ezrahite, as being very wise. In 1 Kings 4: 29-34, Solomon's wisdom is "even greater" than the wisdom of Heman and Ethan. Heman is described as an author, along with the Sons of Korah, of Psalm 88 and Ethan as the author of Psalm 89.

Although Solomon is described as the creator of 1005 songs in 1 Kings 4: 29-34,, only two psalms, 72 and 127 are "of Solomon."  Moses is identified as the author of Psalm 90. Psalm 95 is extensively quoted by the author of Hebrews in Hebrews 4. In Hebrews 4: 7 the author attributes those passages to David.

Among those who consider the Scriptures as inspired by God, there is considerable debate as to what value one puts on the headers of the various psalms.  Are those headers to be given the same authority as the rest of the chapter? Certainly the headers are in all the ancient Hebrew manuscripts and were part of the compilation of the 150 songs into one single document.

Resources

There are thousands of online resources on the Psalms! In addition to those resources described in an earlier post, I've read though some posts at GotQuestions: on the Sons of Korah and on the Psalms of David,