HOLLAND: early 14c., from Dutch Holland, probably Old Dutch holt lant "wood land," describing the district around Dordrecht, the nucleus of Holland. Technically, just one province of the Netherlands, but in English use extended to the whole nation.
NETHERLANDS: from Dutch Nederland, literally "lower land"; said to have been used by the Austrians (who ruled much of the southern part of the Low Countries from 1713 to 1795), by way of contrast to the mountains they knew, but the name is older than this. The Netherlands formerly included Flanders and thus were equivalent geographically and etymologically to the Low Countries.
DUTCH: late 14c., used first of Germans generally, after c. 1600 of Hollanders, from Middle Dutch duutsch, from Old High German duit-isc, corresponding to Old English þeodisc "belonging to the people," used especially of the common language of Germanic people, from þeod "people, race, nation," from Proto-Germanic *theudo "popular, national", from PIE root *teuta- "people" (source also of Old Irish tuoth "people," Old Lithuanian tauta "people," Old Prussian tauto "country," Oscan touto "community").
As a language name, first recorded as Latin theodice, 786 C.E. in correspondence between Charlemagne's court and the Pope, in reference to a synodical conference in Mercia; thus it refers to Old English. First reference to the German language (as opposed to a Germanic one) is two years later. The sense was extended from the language to the people who spoke it (in German, Diutisklant, ancestor of Deutschland, was in use by 13c.).
Sense narrowed to "of the Netherlands" in 17c., after they became a united, independent state and the focus of English attention and rivalry. In Holland, Duits (formerly duitsch) is used of the people of Germany. The Middle English sense survives in Pennsylvania Dutch, name of the people who immigrated from the Rhineland and Switzerland.