What is it? 
"Capture Anything: Save your ideas, things you like, things you hear, and things you see.
Access Anywhere: Evernote works with nearly every computer, phone and mobile device out there.
Find Things Fast: 
Search by keyword, tag or even printed and handwritten text inside images."

Getting started (links to create an account and download the desktop client)

Extensions and Apps
You can access Evernote on any device's browser. But on the devices you use most often, whether computers, tablets, or smartphones, there is probably a client version you can install which will provide the best functionality. 

The Evernote download page also includes a link to install the appropriate browser extension to make it easy to clip and save portions or all of websites to Evernote. 

If you become an Evernote user, Evernote Clearly is a great choice for formatting websites for better reading and saving pages to Evernote.

Ideas for Education
Evernote allows you to create notes and notebooks you can share with other users. It also allows you to publish individual notes and notebooks to a website. These features can make it much more than a personal note-keeping system.

Chronicle of Higher Ed articles on Evernote:
Evernote in higher ed resources from Evernote

Additional Ideas

Worried about security? See How to encrypt Evernote databases and individual notes 

Productivity/Getting Things Done (GTD) ideas for setting up and using Evernote: 

Many tools support sending information out from the tool via email; Evernote provides each user with an email address which can be used to receive information from such applications. One example is the iPad Penultimate app, which enables handwritten notes on the iPad to be sent into your Evernote account.