Yes you can send and receive text messages on your iPad, for free, using one of the apps listed below!
I have downloaded both apps and use them both, they are free, they both work. The free version has some advertising, but for free it doesn't bother me, there is a paid version available for both if you prefer no ads. The person you are texting does not have to have the app installed, you can text anyone, anywhere, locally or internationally, to their cell phone or ipad and yes they can text you back! When you download the app you are given your very own number. The Hey Wire app let me choose from a list of numbers.
Text Plus is about the same as Hey Wire, different set up, different look but they both are essentially the same and offer the same free services as well as an upgrade to paid services.
Give them a try, have fun texting and saving a whole lot of money!