"Die Anfrage der Familie ändert die Beziehungen des Mannes."
"des Mannes" = of the man = the relationships of the man. If you're asking about the ending, check out the links I gave to mjtischer in this thread. Taken from the first link "The genitive form of most neuter and masculine nouns in German is fairly predictable, with an -s or -es ending."
If you're interested, here is the inflection table for "der Mann":
Why, because it's not a German word. Read up on genitive here (the rule for "Familie" is in part 2, the second link):
This link, in case you want it, shows the inflections for "Familie" in all cases:
Hope that helps!