Burp Comparer is a simple tool for performing a comparison (a visual "diff") between any two items of data. In the context of attacking a web application, this requirement will typically arise when you want to quickly identify the differences between two application responses (for example, between two responses received in the course of a Burp Intruder attack, or between responses to a failed login using valid and invalid usernames), or between two application requests (for example, to identify the different request parameters that give rise to different behaviour).
To perform a comparison between items of data, you can paste them into Burp Comparer, or load them from file. However, the easiest method is typically to pass the interesting requests or responses directly to Comparer from any of the other Burp tools:

When items of data have been loaded into Burp Comparer, they appear within the two tables in the main panel:

To perform a comparison, you simply need to select a different item from each table and click one of the "compare" buttons:
You should note that the byte-level comparison is considerably more computationally intensive, and you should normally only employ this option when a word-level comparison has failed to identify the relevant differences in an informative manner.
When you initiate a compare, a new window appears showing the results of the comparison. The title bar of the window indicates the total number of "edits" between the two items. The two main panels show the compared items colourised to indicate each modification, deletion and addition required to transform the first item into the second.

You can select text or hex view of each item. Selecting the "sync views" box will enable you to scroll the two panels simultaneously and so quickly identify the interesting edits in most situations.