If you'd like to create a website that enables users to send SMS messages to mobile phones, there are a few things that you may need, including: an SMS gateway to actually send the messages, knowledge of HTML, CSS, PHP, and maybe JavaScript and MySQL, as well as a place to host the website, like a webhosting service.
In order to actually send the SMS messages from your website to a mobile phone, you may need to use a service that enables you to do this, and depending on what you'd like to do and how many messages you think your website will be sending, you may have to pay for this.
Additionally, you'll need some knowledge of both HTML and CSS to create and style the website. (If you'd like to add some custom graphics, as well, you may want to have some knowledge of graphic design.)
In order to actually submit a form that contains the SMS message and recipient's information, you would need to use a server-side scripting language, like PHP, to actually send the message to the SMS gateway service.
JavaScript (including the jQuery library) would be useful for adding some interactivity to your website, so that users don't have to refresh the page to send a message, etc.
As for the MySQL, depending on whether or not you'd like to allow your users to save messages and view conversations, or if you'd like to log the messages being sent, you would need to store them somewhere, such as in a MySQL database, where you should be able to access the messages using PHP and MySQL code.
PHPAcademy.org actually has free tutorials on how to create an SMS website like this, which you should check out!
http://PHPAcademy.org
SMS Tutorial (Part 1):
http://www.youtube.com/watch?v=ScQXy88Cwm4
SMS Tutorial (Part 2):
http://www.youtube.com/watch?v=vm5Vnsik_xQ
SMS Tutorial (Part 3):
http://www.youtube.com/watch?v=y4Iq3P492gk
Also, the gateway that is used in the above tutorials is txtlocal.com, which is free to try and includes 10 free text messages. (You may have to sign up for a paid version after you've used the 10 free text messages.)
http://www.textlocal.com/
Best of luck and I hope I helped you!