It differs from journal to journal. My experience is that reviewers are not paid, it is done mostly as a scientific courtesy. Specific referees/reviewers are invited from fields of expertise, for instance our lab, an eye development lab, gets papers from other eye development labs as well as other fields of development that involve similar signalling pathways.
It is not a full time job, and the experience is often done in collaboration with graduate students and postdocs in the lab – it is a very valuable skill to be able to read, critically analyze, and decide to support or reject a paper. It means that you 1) must know your background well 2) must be able to process and understand the data 3) decide whether to believe the data 4) decide whether the data agrees with the observations and conclusions 5) find all the shortcomings in the paper 6) know what the journal you are reviewing for is looking for in a paper 7) decide if all of these things make a paper worthy of acceptance. I reviewed a paper for Science – my first pass through, I found the shortcomings, I believed the data, but I didn’t know if it was Science material (even after the shortcomings were addressed). My boss, with more experience than I, was able to look at it and say “This is the first time this type of regulation has been described in this system” which instantly made it Science worthy, so we recommended acceptance with changes. It generally takes at least a few days of full time work to do this, making reviewing a dozen or so papers a year perfectly reasonable (especially with grad student/postdoc help).
The process works as follows. One writes a paper, according to the format of the journal you wish to submit to (for instance here). One submits to an editor (often online), with a cover letter. Usually, one can suggest reviewers to use or to avoid. The editors read the paper and decide to send it out. If there are gaping holes in the paper or the paper should go in a more niche or different journals, it is rejected without review by the editor. Otherwise, it gets sent out to anonymous reviewers (a point of contention – there are lots of debate whether to allow the writers to know who the reviewers are right now) for a month or so, and then it comes back. Usually, it is 3 reviewers. If two recommend acceptance, the paper gets in. Reviews come back, and the editor usually asks for both responses to the criticisms and changes asked for by the reviewers.
Any part of this is debatable. Rejection without review can be appealed. Often times reviewers can ask for revisions that are beyond the scope of the paper and unreasonable. Each point is a debate between the writer and the editor and occasionally the reviewers. If a particular reviewer is way off base with the others, one can request a new reviewer. It is a fairly open, fairly plastic process that allows a collegial back-and-forth between all parties with a minimum amount of political rancor and bickering and a lot of protection of data.
So if you submit a revolutionary paper, your major foci will be addressing why your paper disagrees with previously accepted work and explaining why your work is revolutionary. If you do not do these things appropriately, or your reviewers are unfair, you have opportunity to retune the paper and resubmit, submit elsewhere, or appeal to the editor. If you are unable to do these things, then the review process has worked and your paper should not be published.