Works great on my HTC Incredible (Android) phone.
Yes, books that you read on your pc (or your iPhone) are actually downloaded and can be read offline. On a PC they’ll go into a ‘My Kindle Content’ folder under your Documents directory.
You can even take that file and copy it directly to your Kindle instead of downloading it again over the Kindle’s network. If you get a Kindle, that is. 
The downloading of the file also lets people read Kindle books while unable to connect to the Internet for whatever reason - no WiFi/wireless network access, on a plane, etc. - so this is great for vacations, commuting, waiting rooms, wherever you might want to read a book but may not be able to access the Internet. I read a Kindle book on my iPad on my last plane trip, and saw lots of people with Kindles reading (as well as some people with iPads using some kind of e-book reader).
Kindle - probably the best of the bunch. Good easy to use interface, nice customizations, (and I like the font, so being stuck with it isn’t a problem to me.) Nice navigation choices in particular.
ereader.com - not as slick as Kindle, but pretty good, and has some options that Kindle doesn’t, like ‘paging’ up and down instead of left and right.
ebooks.com reader - fewer options in the interface, but serviceable. Only really works with titles from the ebooks.com website
Bluefire - fairly good interface design, but you can’t override the margins built into the book file itself, which are generally not designed for the iphone screen. That doesn’t leave much screen real estate for words.
iSilo - starting to get to the bottom of the barrel here. Interface doesn’t work too well on iphone, it’s mostly a straight copy of what made more sense with a palmpilot or pocket PC. You can’t page beyond what’s set up as a ‘page’ in the book, (which is normally the entire book or a lengthy chapter,) just scroll through it, and hopping to a point within the page requires going to an options screen. Probably I’ll end up re-converting everything important from this format to Kindle.
Kobo - truly sucky. I don’t have anything from the Kobo store itself; got this because I thought it might solve the margins problem in Bluefire, but compared to this, Bluefire is a very well designed and stable reader.
I’m going to give Diesel a try, and probably Stanza too. Not sure if I’ll ever use iBooks unless there’s something I want that’s only available there.
There is no limit to how many physical devices you can link to your Amazon account. I have 4 PCs, a Kindle Fire, and 4 iphones linked to mine.
The only limit is that, for each ebook title, no more than 5 devices can have that particular title downloaded to be read at any given time.