to bind yourself by a solemn oath, or to put someone else under one
This is the everyday Hebrew word for swearing an oath: calling on God (or some higher power) to hold you to your word, so that breaking it would bring judgment on you. It covers both halves of the act. Most often a person swears, taking the binding on himself. But the same word, in its causative form, means to put someone else under oath, that is, to make a person swear or to formally charge them to do something.
The word does not name a special religious ceremony so much as the weight an oath carries. To swear is to stake your integrity, and ultimately God's name, on what you are promising or affirming. That is why the Bible treats a sworn word as nearly unbreakable, even when it was given to enemies (the deceived oath to the Gibeonites in Joshua 9) or turns out to be costly (Psalm 15:4, the one who keeps an oath even when it hurts).
Its most theologically loaded use is of God himself swearing. Again and again God is said to have sworn the land, descendants, and blessing to the patriarchs, and the whole nation's hope rests on the fact that God cannot go back on his oath. Since there is no one greater for God to swear by, he swears by himself (Genesis 22:16, Isaiah 45:23).
The word is spelled like the number seven, and many have connected the two, perhaps because oaths were sealed with sevenfold acts or sacrifices. The link is widely discussed but uncertain, and it does not change the plain meaning: to take, or impose, a binding oath.