What is the nature of this evil spirit?
The semantic range of both רעה (evil) and רוח (spirit) are quite broad. As WoundedEgo correctly points out, רעה does not always mean evil or wicked in the narrow sense of the word, but can also refer to misery, trouble, disaster, (see HALOT Abridged). The verbal cognate (רעע, be evil) likewise has a broad semantic range. For example, רעע is normally translated 'sad' in Nehemiah 2:3, even in literal translations such as ASV and NASB.
רוח, like spirit in English, is extremely broad in meaning. Some of the possible meanings listed by BDB include breath, wind, spirit, air, disposition, temper, gas, etc.
Thus, considering the semantic range of the words normally translated 'evil spirit', it is completely possible that they do not refer to a personal demonic being that in some way intermittently possessed Saul. That fact that music provided him with relief (albeit only temporarily) would seem argue against such a conclusion.
The fact that רוח can refer to both personal and impersonal forces, such as mental faculties, temperament, etc., perhaps indicates that this distinction was not very acute in the Biblical world view.
As Tony Cartledge comments:
Modern psychologists can easily spot signs of manic depression or
paranoia and other indications of mental illness in Saul’s behavior.
These concepts, however, were unknown to the ancients. Hertzberg
correctly noted that Saul’s suffering was clearly described in
theological terms, not psychological terms. The Old Testament writers
were so opposed to dualism or polytheism that they had little choice
but to assign both evil and temptation to God, who was the source of
all things (e.g., Deut 13:2-4; Amos 3:6; 2 Sam 24:1; 1 Chr 21:1). An
evil spirit between Abimelech and the men of Shechem was attributed to
God (Judg 9:23), as was the lying spirit that came over the false
prophets who opposed Micaiah (1 Kgs 22:19-22).
This insight has much to commend itself, although the author's statement that "The Old Testament writers...had little choice but to assign both evil and temptation to God" seems to imply that they were mistaken in doing so. Perhaps it is our modern categorizations that are out of sync with reality, and that the distinctions between theological and psychological, physiological and spiritual are not as clear cut as we think.
How can an evil spirit be from God?
Whether "evil spirit" does refer to a wicked demon or not, the question remains as to how God could possibly be responsible for sending something bad, sad, or troublesome, regardless if it is a spirit, gas, or a disposition.
Even a cursory reading of scripture should make it clear that God regularly brings what man considers evil, disastrous, calamitous, in order to judge, to bring man to repentance, or even to bless.
If one is inclined to understand the 'evil spirit' in 1 Samuel 16:14 as referring to some kind of demonic being, this is by no means out of harmony with the rest of scripture. The locusts in Revelation 9, which many understand to represent demons, explicitly receive their power to torment from God.
This does not mean that God is the ultimate source of evil, but that as sovereign, he is completely capable of using evil to achieve his purposes. As the NIV Zondervan Study Bible notes:
“The Lord does not perform evil, but evil elements are under his
command in order to bring about his purposes (Judg 9:23; 1 Kgs
22:19–22). The sovereignty of God is such that everything that happens
in heaven and on earth are under his divine control.”