TextMe is a free service that works using your existing internet data plan (the same one you use for email and web browsing).
You need to have an internet connection to use TextMe.
You still have to check your phone operator contract for your data plan details, but if you have unlimited internet access from your smartphone, you won't have any extra charges.
You can also use WiFi.
TextMe works on WiFi, 3G or 4G or Edge connections, whichever is available.