A first impression is the first print run of an edition: every copy printed in the first batch from the first setting of type. British publishers have traditionally used "impression" where Americans say "printing", so "first impression" and "first printing" mean the same thing.
How to spot it
On the copyright page, a later impression usually announces itself: "Second impression", "Reprinted 1956" or a list of reprint dates. A first impression has no such line. Where there is a number line, the lowest number shows the impression, so a line that still includes a 1 is a first. Some publishers marked first impressions explicitly; others simply left later impressions to be labelled.
What it does to value
For collectors, the first impression is usually the copy that matters. A first edition that turns out to be a second impression is generally worth far less, even though the text and binding look the same. The difference is largest for books that sold slowly at first and then became famous, because the first print run was small.
See first edition vs first printing for how the terms relate, and the number line decoder to read the copyright page.