During the controversy of Galileo and the Church, the actual point of contention was this very work which Galileo published.
In the Dialogue, he was supposed to set forth arguments for and agains the Ptolemaic worldview (the unmoving earth in the centre of the universe) and the Copernican (the earth and other planets going around the sun). This book does that, and brilliantly, showing Galileo's resourcefulness as a scientist, philosopher (at least to an extent!) and writer. The charge against him was that rather than being even-handed, the book was clear support of Copernicanism. This is a non-obvious topic but what is obvious is the
importance and magnificence of the work in terms of both the subject
matter (the importance of the structure of the universe) and method (a colourful dialogue containing heated debate which spans literally dozens of arguments for and against each system).
The work has 3 characters: Salviati who is a Copernican, Simplicio who is an Aristotelian and follower of the Ptolemaic system, and Sagredo, a non-affiliated but intelligent person. They meet and debate over 4 days. The first deals with the question of whether the substance of the heavens is fundamentally different to the earth as well as some other fundamental assertions of Aristotelianism. The second deals with the earth's daily rotation. The third is about the alleged yearly orbit of the earth around the sun. The fourth (considered by Galileo to be the crown of his argument - which is all the more endearing as it is wrong) is about the cause of the tides.
Reading this is especially interesting because all of us believe that the earth goes around the sun, so it's easy to just approach this simplistically. But the reality is, it was an actual matter of debate, as the book shows. As per the edition, it contains a very good, readable translation along with Galileo's margin notes and good footnotes which unfortunately aren't matched to the body text so you have to flip forward and back.