A unique ID (or UID) is a unique integer number of a page or content element that is used internally by a TYPO3 system.
When an element (i.e. database record) is created in TYPO3, it gets such a unique number assigned. That number will never change. The UIDs are unique for each type of record (pages, content elements, extension data records). All relations between elements (e.g. which content elements belong to a certain page) are handled by the internal IDs. The main advantage compared to relations that use names is that names can change over time and the relations would be no longer valid.
An editor usually isn't aware of the IDs unless she wants to place an internal hyperlink to another page within a text content element. To do so, he uses the UID of the target page in the TYPO3 link tag:
<link 42>anchor text</link> (Here, '42' is the ID of the page to be linked).
How to display the IDs of TYPO3 page in the page tree is shown in our screencast Displaying IDs in the page tree (in German).