here. Basically the feature builds on the existing "Skills" facility, allowing connections viewing your profile to click on a skill to indicate that they can confirm that you have that skill. This is very similar to LinkedIn Recommendations, but there are some differences, which are crucial.
The problem with Recommendations is that they're time consuming to write (properly) and also need a level of (necessary) interaction between the two parties before being published against a profile, nobody wants a Recommendation to say negative things, so each one must be approved by the recipient. Endorsements, however, are quick and precise - you can select the skills you've seen your contact demonstrate and endorse each with a single click.
Also, Endorsements are connected to Skills, and everyone should have Skills listed on their LinkedIn profile, because having a formalised skills list makes it easier to be found on LinkedIn searches. I've seen numerous LinkedIn profiles which have not been updated to include Skills, and the Endorsements feature will encourage those people to add Skills, so that they can then be endorsed.
As mentioned above, Recommendations take a little while to sort out, but they can be very specific, and allow your connections to go into a bit more detail about the particular business or technical projects you've worked on, and how significant your contribution was. All of that is lost in a single-click Endorsement.
The other major issue with Endorsements is the amount of noise they can generate. You will receive emails from LinkedIn whenever someone endorses your skills, but you can switch these off (go to Settings -> Email Preferences -> Set the frequency of emails -> Endorsements). Also, whenever someone in your network receives or issues an Endorsement, that activity will show up on as an update on your LinkedIn home page - although LinkedIn will roll these up into single updates for multiple connections from time to time.
The speed advantage offered by Endorsements can work against you, however. If one of your connections gets click happy, you may find them endorsing you for skills that they have no knowledge of. Your network will consist of people you have, and have not, worked with before, so it's worth keeping track of those Endorsements - if any are not appropriate, you can hide them (click the "see all endorsers" icon, and use the "Hide endorsement" buttons as required).
And if you are a "click happy endorser", remember that your Endorsements will show up as a LinkedIn activity, so if people in your network see that you're endorsing pretty much everyone for pretty much everything, your Endorsements will carry less weight in the future.