If you are having odd problems with your iOS device, there are some basic steps you can take first that might resolve the issue. First is to simply reboot the iPhone by holding down the power button and then "slide to power off." Then hold the power button again until it starts up with the Apple logo.
Next you can try a hard reset. You can do this by holding down the on/off button and the home button until the Apple logo appears.
Another thing to try is resetting all of the iOS device's settings. Go to Settings > General > Reset > Reset All Settings. While this option should keep your data intact, it's always a good idea to first do a backup. If resetting all settings doesn't go far enough, you can choose Erase All Content and Settings, and this will wipe all data and put the phone back to its original configuration.
If you can't even get to the Settings app to begin with, you'll need to plug it into a Mac or PC with iTunes and follow these steps from iMore.com.
If putting the device into recovery mode still doesn't get things working, you can put it into DFU mode. DFU stands for "Device Firmware Update." This will require plugging your iPhone into a Mac or Windows PC with iTunes running. It's all about the timing of it, and it's easy to get it wrong, so it may take more than one attempt. Putting the phone into DFU mode is a complicated process, so you'll want to check out this article from iMore for step-by-step instructions on how to do it.