What does "square the base" mean?
The controller wants the base to have two distinct 90° turns, from downwind to base, and base to final. And not one continuous turn from downwind to final. And no shortcuts to be taken to shorten the downwind.

It means the base will be slightly longer, and the downwind more parallel to the runway.

The controller is buying time (delaying the arrival) to accommodate someone else.



Top image is square-the-base, as opposed to the tight circular pattern—which experienced pilots tend to prefer1. (Images above are a mere representation.)

Note: exact answer will depend on the plane's exact location in the pattern.

It's one of those terms you won't find in a manual, like "hold our code".

It's a shorter time extension than "extend downwind leg":





Further reading: studentpilot.com, AOPA, 1 humorous AVweb video.