Do you have </form> and the end of the form, to end the tag? If you do, then theres gotta be something wrong with the process page.
By the looks of it, your using a pre-made script, maybe the script is from an old PHP version, older than the one your server supports, or maybe you don't have PHP at all. Other than that, I'm not sure. The only thing worng with what you showed is there's no </form> at the end. Plus, it could be sending it, but it's in your bulk or spam folder, PHP mail tends to do that sometimes.