The Psalms are always referred to as The Psalms of David. It is said that Moses gave the five books of the Pentateuch to Israel and that the five books of Psalms were given by David.
The Psalms reflect human experience with emphasis on the hardship of life for people who fear G-d and their struggle to abide by the laws in spite of the oppression and success in life of sinners.
The main reason the Psalms appeal to a wide range of people throughout time can be well understood when we realize that this struggle is not confined to a group of people or to a period of time.