In modern usage, curse is generally considered synonymous with swear.
In the Bible curse takes on different meanings, such as to invoke misfortune, to afflict, to make an oath, and to insult, as well as to swear. Here are some ways curse is used in the Bible:
Curses, or ill-will, can be given to people (Canaan in Genesis 9:25), the disobedient (Jeremiah 11:3), a house (Job 5:3), animals (the serpent in Genesis 3:14), plants (the fig tree in Mark 11:12-14), and earth (Isaiah 24:6); the ground (Genesis 3:17); and the land (Hebrews 6:8).
Curse can also mean “to swear at” (Goliath cursed David), 1 Samuel 17:43.