geen = none, no amount of, not any. (It is like the German word kein.) In this example it is like "I don't eat any dinner" or "I am not eating any dinner"
geen is also used in some situations where "not" would normally be used in English.
not = niet, geen
nee = no. The opposite of yes.
Well, the Dutch version is already here as the topic of this thread. "Ik eet geen avondeten." In German it is "Ich esse kein Abendessen"
These single sentences can translate to several different sentences in English.
1) Do not = Don't. - 'I do not eat any dinner.' / 'I don't eat any dinner.'
2) The 'any' is optional. Geen means 'not any' / 'none' / 'no'. In English this is already covered by the negation of the verb. - 'I do not eat dinner.' / 'I don't eat dinner.'
3) Can also be written in present continous tense - 'I am not eating any dinner.' / 'I am not eating dinner.'
4) I am = I'm - 'I'm not eating any dinner.' / 'I'm not eating dinner.'
5) The noun can be negated instead of the verb. - 'I eat no dinner.' / 'I am eating no dinner' / 'I'm eating no dinner'
Some of these have identical meanings, but between others there are quite large differences in meaning - eg. talking about the present moment vs. in general. - However, I'm pretty sure all of these eleven English sentences are correct possible translations of the single Dutch and German sentences.
That is definitely wrong :) the translation is: Ik eet geen avondeten. Ich esse kein Abendessen. But you are right that german and dutch are quite similar, much more similar than dutch and english. Like the order of the words in a sentence, that is almost a 100% match between dutch and german.
The word "any" is of course the problem. The difference between "no dinner" and "not any dinner" is a small difference. "no dinner" is that in a month you would have dinner 0 times, which is approximately the same amount if you say "not any dinner", but with "no dinner" exceptions are possible. "No dinner" except for special occasions. "I don't eat any dinner" would translate to: Ik eet TOTAAL geen avondeten.