When you receive a text message on your iPhone your phone will vibrate, a preview of the message will display and you'll hear an alert chime (if sound is turned on). If you choose to ignore the alert, your iPhone will remind you of the message two more times - again, with vibration, preview and alert chime.
While three notifications certainly helps make sure you're aware of incoming text messages, one alert might be enough for you. You can limit text message notifications to just a single alert in Settings.
Launch Settings. Scroll down and tap Messages. On the next screen, you'll see (amongst other things) a "Repeat Alert" setting. By default this is turned on. Tap the switch to set it to the OFF position.
Now, when you get a text, you'll only be notified once.