I've sent messages through the website before too. I have Verizon and I used to have a $5 unlimited text messaging to other Verizon users, as most of my friends did. But when I started work again I didn't send nearly as many as I did over the summer. So I ditched the plan and I use the website. That way I'm not paying for the text message, and neither are my friends.
I also used the website when I had Verizon and my friend has Sprint -- I went to the Sprint website to text message her. You can also enter your email address as the call back number so you don't receive replies to your cell.
Likewise, you can send text messages from any email account. It would vary on the cell phone carrier but you would send a brief email to the area code + cell phone number (10 digits) i.e.
5851234567@vtext.com (This would be sent directly to a verizon cell phone).
Remember though, that text messages usually have a character limit of around 160.