The chompSMS application is free. It's available on Google Play™, and you install it directly on your phone. Easy!
Or, if you don't have access to Google Play, you can install chompSMS directly from your browser on your Android phone.
Once installed, you can start using chompSMS straight away as you would the built-in SMS application.
To see how SMS messages are sent, look at the send button in the chompSMS app:
Just like sending using the built-in Messaging app, using your mobile carrier.