Hehe, good question, next question?
To be honest I had to look it up myself.
Here are some rules:
The genitive case requires the ending „s“ or „es“ for masculine und neuter nouns. Feminine nouns stay as they are. Your example “die Regierung” is feminine -> no “s” but “das Mädchen” is neuter (how crazy that may ever sound) so the genitive is “des Mädchens”.
“Den Kindern” is dative and plural though, therefore it requires the “n” ending.
I found these rules on the website: http://easy-deutsch.de/en/nouns/
Perhaps you want to have a look at it because there are much more rules to consider and I found it’s all pretty well explained there.