Still the limb, and much besides
In ordinary English, a hand is the grasping end of the arm: palm, fingers, and thumb. That physical sense is the oldest and still the first that speakers reach for. Around it sit later jobs. To give someone a hand is to help. To have a matter in hand is to control it. A farm hand is a worker; a sailor can be one of the hands. Cards come in a hand, a clock has hands, and an audience gives a hand. Almost all of these grow from two early ideas already present in Old English: the limb that seizes, and the side of the body that limb occupies.
Inherited, not borrowed
This is a native English word. It did not arrive with Norman French, and it is not a later borrowing from Latin or Greek. Old English already had hand and the variant hond. Both named the human hand. Both were also used for side or direction, especially right versus left, and for power, control, or possession, by a short step from grip to authority. The noun is feminine in Old English and belongs to the small class of u-stem nouns, the same morphological type as duru (door).
The Old English plural was handa. That form did not survive. Middle English first replaced it with handen, then with the regular plural hands that English still uses. Middle English spelling is restless: hond, honde, hand, haund, and a few others. Some forms with a front vowel, such as hend, point to contact with Old Norse, which had hǫnd and a dative hendi. The main line of the modern word continues the native Old English noun rather than the Norse one.
The source is Proto-Germanic *handuz, reconstructed because the same item appears, with regular sound changes, across the Germanic family. German has Hand, Dutch hand, Swedish hand, Danish hånd, Icelandic hönd, West Frisian hân, and Gothic handus. The match is close enough that a speaker of one of these languages still recognizes the others.
Where the trail stops
Beyond Germanic the etymology is unfinished. Comparative linguists reconstruct *handuz with confidence; they do not agree on what, if anything, stood behind it in Proto-Indo-European. That uncertainty is the scholarly position.
Several proposals are in print, and they should stay separate from the facts. One derives the noun from a pre-Germanic form related to the strong verb *hinþaną, to reach for or obtain, so that the hand would be named as the part that takes. Older dictionaries already compared Gothic hinþan, to seize, and some accounts join the same cluster to English hunt. Another suggestion starts from a Proto-Indo-European noun for the hand, sometimes written *ḱómt, and even tries to connect that noun with the word for ten, on the idea that ten is two hands of five. Elmar Seebold has linked the Germanic word to the Greek family of κεντέω, to stab. A further view treats *handuz as a substrate item taken into early Germanic.
None of these is settled. A connection to the Indo-European root *ǵʰend-, seize or take, is attractive in meaning and difficult in sound: the expected Germanic outcome does not fall out cleanly. What can be said without hedging is negative and useful. English hand is not descended from Latin manus, the source of manual, manuscript, and manage. It is not descended from Greek kheír, the source of chiropractic and, through a longer path, surgery. Those are parallel names for the same body part, not ancestors of the English word.
Old English even kept a rival. Mund, related to the Latin manus family, could mean hand, but more often protection, guardianship, or a protector. English kept hand for the limb and let mund fall out of ordinary use. The pattern is typical of Indo-European. Languages in the family do not share one ancient word for the hand. Many terms come from verbs of seizing or collecting, or from a part of the hand rather than the whole. Irish lám and Welsh llaw go with Latin palma and originally meant the palm. Germanic simply used a different label and then held onto it.
How the senses multiplied
The body-part meaning never left. Side and power are already Old English. By about 1200 the word appears in language about pledges made with the hands; by the late fourteenth century it can mean a promise of marriage, and also handwriting or a person's style of penmanship. First hand and second hand, from the mid-fifteenth century, imagine an object passed from palm to palm. At hand is near in time by about 1200 and within reach by about 1300. On the one hand and on the other hand, recorded from the 1630s, take the old left-and-right sense and turn it into a way of weighing an argument.
The verb is later than the noun. Around 1400, to hand meant to seize or take charge. The everyday sense, to pass something to someone, is from the 1640s. Earlier English already had other verbs for related actions, including hend, from Old English gehendan, and handle. Handle and later handy sit on the same Germanic stem. Handsome, attested in the fifteenth century, first meant easy to handle; the sense attractive is a later development.
Trade and measurement added still more. From the 1560s a hand is a unit of length, first three inches and then four, now used almost only for the height of horses. From the 1570s it is the pointer on a clock or watch. From the 1580s it is a person who works with the hands, then a hired laborer (1630s) and a member of a ship's crew (1660s). Playing cards held by one player, and a round of play, are from the 1620s. A round of applause is later still, from 1838.
None of these later senses needs a separate etymology. They are the same inherited noun, stretched along lines that Old English had already opened: grip, work, side, and the visible gesture.
A short end to a long word
The origin of the word hand is short on the far side of Germanic and crowded on the near side. English did not import it. Speakers inherited a common Germanic noun whose deeper source remains disputed, then spent more than a thousand years hanging extra jobs on a name that still, first of all, means the thing at the end of the arm. If a single older form is worth keeping in mind, it is Gothic handus: already the same word, written down centuries before Middle English settled on hands.