Great question!
The correct name of the nation is 'The Netherlands'. This comprises the whole country and its 12 provinces.
The name Holland is originally used exclusively for the two provinces in which most of our major cities are, North-Holland & South-Holland. However, more recently people use both Holland and the Netherlands exchangeably and we aren't mad. In fact, our company's name is based off of the easier way to say it. We don't like to fuss.
When something is from the Netherlands, it is Dutch. The Dutch language, Dutch weather and yes, even going Dutch is a thing here. This is a bit confusing, since the German demonym is 'Deutsch'. We aren't Deutsch, though. Keep it Dutch and you will enjoy it very much!