In that case, the process will probably work a little like this.
Your agent rounds up a possible 8-12 editors. That’ll mostly be editors who work at large publishers (though often in semi-autonomous imprints or companies), but there’ll be 2-3 smaller independent publishers, as well, more than likely. Your agent will introduce editors to the book, check they’re available (not off on holiday, etc.), then more or less simultaneously get the book to them. (Books used to be sent on paper. These days, it’s often electronic.)
You wait. Your agent will be looking for a first offer. As soon as they get an offer:
They’ll start calling everyone on the list, setting deadlines, coaxing offers, etc. A book auction chemistry is critical and delicate. Get three rival offers from three rival publishers, and you should do well, except many notionally independent publishers are connected (e.g. Transworld, Orion, Hodder, Headline), and these guys don’t bid against each other. A smaller publisher (Quercus, Faber, Profile, Atlantic) may be a wonderful publisher, but they won’t be able to fight the bigger ones on advances, so a financial outcome does depend very much on where interest lies.
Then your agent will call for ‘best and final’ bids, then close a deal.
A contract may take a while to follow. I’ve known it take as long as 6 months, but a verbal agreement is nevertheless something you can depend on. These agreements should never sour.
So much for your home nation deal (i.e. the UK if you’re British, US if you’re American). Your agent will then start to target major overseas markets. Most agencies will have someone in charge of foreign rights, who’ll be talking to publishers or sub-agents in Germany, Japan, France, Italy, Spain.
Your agent will also have a sub-agent in the US (if she’s British) or the UK (if she’s not). That sub-agent will be also start sending your book around. Note it’s often easier for US authors to get a UK deal, and harder the other way around. What’s more, books that seem obviously US-friendly are oftener ones that make no impact. Ones that seem obscurely British or quirky often do well (Nick Hornby’s Fever Pitch is a famous example).
And once those major markets have been dealt with, your agency’s attention will start to turn to other areas where small but meaningful deals can be done. India and China, by the way, may well buy your book, but mightn’t do so for much money. I
This entire process can easily take about a year, perhaps more.
And by the time the last paperback publication advance drops into your account, you’re quite likely not just onto your next book, but the one after that.