Out of context, this verse can be interpreted a number of ways. But in the context of the Psalms of David, "the enemy" is used often and refers, not to Satan, but to particular enemies that either David or the nation of Israel was up against at the time.
Looking directly either side of this Psalm, "the enemy" in Psalm 7 is described in the introduction as "Cush, a Benjaminite":
if I have requited my friend with evil, or plundered my enemy without
cause, let the enemy pursue me and overtake me, and let him trample my
life to the ground, and lay my soul in the dust. Psalm 7:4-5
In Psalm 9, however, "the enemy" refers to nations laid at the feet of Israel:
Thou hast rebuked the nations, thou hast destroyed the wicked; thou
hast blotted out their name for ever and ever. The enemy have vanished
in everlasting ruins; their cities thou hast rooted out; the every
memory of them has perished. Psalm 9:5-6
In Psalm 106, "the enemy" refers specifically to the Egyptians who pursued the Hebrew people to the Red Sea:
He rebuked the Red Sea, and it became dry; and he led them through the
deep as though a desert. So he saved them from the hand of the foe,
and delivered them from the power of the enemy. And the waters covered
their adversaries; not one of them was left. Psalm 106:9-11
In Psalm 74, "the enemy" is also "thy foes" - an army that has defeated Israel, and then attacked and desecrated their holy places. This is as much an attack on God as it is on the nation of Israel, and the enemy is portrayed not just as Israel's enemy but also God's.
Direct thy steps to the perpetual ruins; the enemy has destroyed
everything in the sanctuary! Thy foes have roared in the midst of thy
holy place; they set up their own signs for signs.... ...They set thy
sanctuary on fire; to the ground they desecrated the dwelling place of
thy name. They said to themselves, "We will utterly subdue them"; they
burned all the meeting places of God in the land.... ...How long, O
God, is the foe to scoff? Is the enemy to revile thy name for ever?
Psalm 74:3-4, 7-8, 10
This is not a reference to Satan, but those of Israel's enemies who, by their actions, position themselves also against God.
"The avenger" is used in Numbers to describe the man who seeks revenge against someone who has killed another man (unintentionally or not).
The cities shall be for you a refuge from the avenger, that the
manslayer may not die until he stands before the congregation for
judgment. Numbers 35:16
The "avenger of blood" is apparently entitled to kill this manslayer as soon as he meets him.
The enemy and the avenger
The complete phrase "the enemy and the avenger" is also used in one other Psalm:
All day long my disgrace is before me, and shame has covered my face,
at the words of the taunters and revilers, at the sight of the enemy
and the avenger. All this has come upon us, though we have not
forgotten thee, or been false to thy covenant. Psalm 44:15-17
It appears here to be a general phrase that encompasses 'all those who would seek to harm me' - physically, as opposed to "the taunters and revilers" who would do so verbally.