Suppose that you go to a party where everyone knows at least one other person, you get a bit bored and wonder whether there are at least two people which know the same number of people there.

To get our head around a problem like this is always very useful to think of a few examples. First supposed that this party is a bit borining and there are only 2 people, then by assumption they know each other so this is easy. Now suppose that there are 3 people, and what are the possibilities for the people they know? well they either know 1 person or 2 people, but there are 3 of them so by the pigeonhole principle there must be some overlapping, i.e. two people know the same amount of people. 

Ok this is looking like there will always be at least two people which know the same amount of people in the party, can we prove it for a general party of n people? yeah! reasoning as above there are n people but the amount of people that they know is either 1,2... n-1 so by the pigeon principle there must be some overlapping. 

Good, now that you have sattled this worry you can  go and enjoy your party.