I use it for a combination of things, but mostly as a place to put and file things I might want later.
I store articles I find on the net, and download the full-version of them, with the Evernote Web clipper. This replaces Pocket, and other apps, and allows me to have synced versions on my phone.
I store Dungeons and Dragons Gamenotes, and have thema cross whatever laptop I happen to bring to game. I GM, so there is a lot to keep track of.
I keep recipes that I yanked from my Mom and Grandmother, so I can type in "cookies" and quickly find how to make them.
Basically, I use it for any content that I want to save, sync, and search.
Could I use The Filesystem+Spotlight+Dropbox? Probably. But the finder's UI has to do a lot. This has a UI which is dedicated to my use-cases. For me, it makes a lot of sense.
That's what I use currently.
Also, I could not find any integration with kindles, except for 'open the website in your kindle'.
Evernote is better than the file system/Dropbox primarily because it does full text search of PDFs (even scans), words in photos, and plain text.
Evernote also lets you look at all file types in the same interface. And there are good mobile apps.
It's sort of like an integrated version of Spotlight, Quick View, and Dropbox but with better search and better mobile apps.
Every online receipt or payment confirmation, flight details, etc gets inserted into evernote via OS X's "Save PDF to Evernote" link in the print dialog.
I hardly ever look at the stuff, but it's nice to have it all indexed and immediately searchable.
Yojimbo did, and it was nice ;)