How to add SMS text messaging functionality to my website?

I believe that the new industry leader is Twilio, they have an excellent API and reasonable prices. Some alternatives worth considering are Tropo, Plivo, and Nexmo.

You can also use an SMS Gateway like ClickATell.

Best of luck!

EDIT: I found a question on StackOverflow that has lots more suggestions, some may be outdated.

I had a friend who managed to implement this via email. I believe he did it exactly the same way this site recommends:

Virgin Mobile:
AT&T: [email protected]

where phonenumber = your 10 digit phone number

I've been using Twilio for voice and I've been quite impressed. I'm sure their SMS system isn't much different. This will allow you to send to any carrier without knowing the email address to send to (as I've noticed they change over time).