Anna,
Yes, Flemish is the adjective form of Flanders and that is currently a region in Belgium. Until 1830 (part of) Belgium was part of the Netherlands, but they rebelled/seceded. Thus, the 16th and 17th Flemish painters were Dutch in that era (although many consider them Belgian now).
As for the distinction North and South Holland, it does not refer to the total country. Of the 12 provinces we currently have, one is called North Holland and one is called South Holland. Together they form the Western part of the Netherlands (North, South, West, are you still there?
), excluding the Southwestern part of the Netherlands. So when people refer to the Netherlands as Holland they're actually referring to just a part of the country (it would be a bit like referring to the USA as Virginia).
The Lowlands is more or less a nickname for the total Dutch/Belgian region. Because the country IS flat. It's kinda like referring to Australia as Down Under. BTW, the French (Pays Bas) and Spanish (Paisos Bajos) names for the Netherlands literally translate to Low Countries.