When I first tried to understand composite design pattern, I came across this UML explanation
Image source : wikipedia
To start with, I was not able to make any sense out of this, so I tried to look for simpler example, and came up with this
Think of a scenario where we need to support a tree format data structure. Where every node can be composed of more similar nodes, or a leaf node. For example in above scenario, at leaf level we have juniors, to whom no one report. At non-leaf level we have managers, to whom, juniors and other managers report, and these managers themselves report to some senior manager. To help us structure this kind of complex relationship, composite design pattern comes into play.