Dutch is a Germanic language which sits somewhere between English and German. The grammar is something like German with the worst complexities removed - no cases, regular plurals and gender is only animate or inanimate. A lot of vocabulary is similar to German, eg "mogelijkheid" means possibility (German "Möglichkeit"). Other vocabulary does not resemble either language, eg "fiets" for bicycle (German "Fahrrad"). The High German consonant shift did not affect Dutch so better is "beter" (German "besser") and hope is "hopen" (German "hoffen"). One of my favourite Dutch phrases is "een kindje kopen". Literally it means to buy a child, but the real meaning is to have a child /give birth. If only it were that easy!
Standard written Dutch developed in Amsterdam in the 1600s, but it had considerable southern influence because of the large number of refugees and the high status of the Flemish merchants. The standard language is now called AN ("Algemeen Nederlands") but the name I'm used to is ABN or "Algemeen Beschaafd Nederlands" (=General Civilised Dutch). The Netherlands, Belgium and Suriname together form the "Nederlandse Taalunie" (Dutch Language Union) which promotes the standard language.
Flemish is the name of the variety of Dutch which is spoken in Flanders. I liken the difference between Flemish and the Dutch of the Netherlands to the difference between British and American English - distinctive accents, some differences in vocabulary and a few differences of grammar. Flanders also has strong local dialects - some people say that they cannot understand people from the next village.
I'm pleased to find that I can still understand a lot, including a fair bit of the radio news. People never look surprised when I speak Flemish, yet it must be relatively rare to hear it spoken by non-natives. Or maybe they just think that I come from a different dialect region. And no-one replies in English.
In certain respects I learnt Flemish like a native speaker - I picked it up through listening and conversing. I don't know the grammar explicitly - I could never explain why something is correct or not - it just sounds right to me, or not. That's quite different from other languages which I've learnt at school or as an adult. When I was in Belgium, I didn't speak English at all for 8 months from January to August. I wrote my diary and dreamt in Flemish. By the time I came home, I was actually speaking English with a foreign accent. For several years afterwards, a few words came more naturally in Dutch than in English.