"O'Donnell" is an Anglicization of the Irish-Gaelic surname "Ó Domhnaill."
"Ó" literally means 'grandson (of)' but generically in means 'descendant (of).'
"Domhnall" is an ancient given-name believed to mean "world-ruler." It has long been equivicated to the Biblical name "Daniel" due solely to both names having a d-n-l consonant combination.
The reformation of the Irish-Gaelic language during the Gaelic Revival allowed for the simplification of many names, and "Domhnall" is now far more commonly rendered as "Dónall." Though the hint that indicates the name's origin is lost, pronunciation in a country dominated by English is much simpler. This extends to the surnames so derived, and "Ó Domhnaill" is frequently rendered as "Ó Dónaill" modernly. (Please note that "Dónall" spelled without the accent or with only one 'L' is incorrect or at least very odd in Gaelic. These are more often English spelling variants.)
There are a few major "O'Donnell/Ó Domhnaill" family septs (a 'sept' is like a Scottish 'clan'). Most prominent is that of Tirconnell (Co Donegal). Other major families with the name are those of Thomond (Co Clare) and Ui Maine (Co Galway).