Imagine A and B are trying to have a conversation. C wants to run a denial of service (DDOS)attack against A so he simply talks to A constantly to the point of B not being able to have her conversation with A. So this is easily stopped if A starts ignoring C and he can have a regular conversation with B.
C, wants a better method to ruin any conversations that A has. So he recruits D and E to help . C, D and E now continuously talk at A. This is a distributed denial of service attack. If A was speaking to large volumes of people it would be very difficult to determine who just wanted to speak legitimately.
So relating that to a real world example, A could be a website and everyone else could be requesting a web page. Given enough load the web server will become slower and may crash - thus causing nothing to be served.