Abarim Publications' online Biblical Greek Dictionary
The verb ομνυμι (omnimu), also spelled ομνυω (omnuo), means to swear (an oath), and swearing is a peculiar thing. Swearing was invented when people discovered that language could be used for various degrees of seriousness, and even for blatant lying. Our verb ομνυμι (omnimu) stems from the Proto-Indo-European root "hemh-", which means either to swear or it means to passionately grab hold of and pull toward oneself (hence, possibly, the Latin derivative amare, from whence the familiar amor, love).
With swearing, one would marry oneself to one's statement, and so formally and solemnly vow to the fidelity of it. All that drama would not be necessary, of course, if one were widely known to never utter a falsehood, which is probably why Jesus urged to not swear and let one's yes be yes and one's no be no (Matthew 5:33-37, also see James 5:12). Alternatively (as the fictional Mark Zuckerberg character submitted in The Social Network, 2010), if one would speak the truth only when accompanied by vows and oaths, one could just as easily succumb to lying under one's oaths. One is a liar always or never, but not sometimes (and that solves the ancient riddle of the liar who confessed to lying: such a man is like dry water and cannot exist; said otherwise, a liar who tells the truth is still a liar, and his truth exists only relative to his earlier lies, which makes even his truth a lie).
The crucial failure of humans swearing, however, comes not from their intent to lie but their inevitable incomplete knowledge and power. No human always possesses all relevant knowledge, and no human controls any situation wholly (or at all, actually). That means that not even a person's most grandiose swearings perfectly guarantee the validity of that person's reports of past events or fortellings of future ones. You never know what you don't know, and you never know what might happen next. Only God does, which is why only God can swear (Luke 1:73, Acts 2:30, 7:17, Hebrews 3:11, 3:18, 4:3, 6:13, 7:21).
When God swears, he indicates that he won't change his mind, which he does, at times (Genesis 6:6, Exodus 32:14, Jeremiah 26:19, Amos 7:3). This difficult concept indicates that God, though omniscient, reacts to reality as it develops, and that quantum mechanics indeed is correct in stating that speed and position of a particle cannot be known simultaneously, not even by an omnipotent being, because these qualities do not exist as discrete entities, not even to God (who, per his own nature, doesn't bother with things that aren't real or true). When God swears, he indicates that what he swears about is governed by natural law, and natural law never changes or allows exceptions (Matthew 5:18).
Our verb is used 27 times; see full concordance, and, notably, has no derivatives. The word for oath is ορκος (horkos).