"He does not like the families."
Translation:Er mag die Familien nicht.
You're onto something. That example sentence would have been better as "Ich trinke meine Limonade nicht". One of the confusing things about "nicht" is that it can negate either an entire sentence or just a part of it, depending on whether it is placed at the end or after the verb. While "Er mag nicht die Familien" might be said under some rare circumstance (emphasis on not liking families but perhaps liking something else) it is awkward. We try to teach the most natural word order where we can. Maybe this resource makes more sense: http://www.canoo.net/services/OnlineGrammar/Satz/Negation/Kontrastierend.html
This is the site I used: http://www.class.uh.edu/mcl/fll/Germ/order.html#Position of ....and this is the site I used to remember what all the silly english grammar terms meant :P http://www.chompchomp.com/terms/
If I understand correctly, nicht essentially goes at the end here because mag is an action verb, not a linking verb. If it had been linking verb, nicht would have gone before the object.
Action verb (like): "He does not like the cat" - "Er mag die Katze nicht"
Linking verb (is) "He is not the cat" - "Er ist nicht die Katze"
There are of course other rules for other situations. (Couldn't be entirely simple, right?)
Annoyingly that directly conflicts with the example sentence you pointed out from about.com. "Trinke" is definitely an action verb in that sentence. However, since that site seems to conflict itself ("Ich trinke nicht meine Limonade" is a declarative statement if ever I've heard one :P), for now I'm going with the example sentence being wrong. Could be there's a rule I don't know about that makes me wrong, but if there is I don't know about it.
If a fluent German speaker would like to correct me, please do. This is a pretty frustrating subject and I'd strongly like to understand it better.