So your iPhone is stuck in a Recovery Mode? (It's seemingly frozen or stuck in a loop, displaying an iTunes symbol with a plug-in cable pointing towards it).
I go on the presumption that your device went into Recovery Mode because an iOS update didn't work out.
But before I go on, this question needs to be asked: Did your device's hardware get damaged? For example, if it tumbled down a flight of stairs and the glass got cracked, or the housing suffered dents. Or ...did it get wet? If it did sustain damage, contact Apple Support or take it to a repair tech. Look up a certified Apple Repair Tech in the White Pages or run an online search for a reliable Tech in your area that knows how to fix Apple products.
I'll give you the potentially bad news, first. Quoting from payetteforward.com : "DFU stands for Default Firmware Update, and it’s the deepest type of restore you can do on an iPhone ...If your iPhone is damaged in any way, and especially if it’s water-damaged, a DFU restore may break your iPhone. I’ve worked with customers who tried to restore their iPhones to fix a minor problem, but water had damaged another component that prevented the restore from completing. A useable iPhone with minor problems can become completely unusable if a DFU restore fails because of water-damage."
But if your device didn't sustain such grievous damage, let us begin the recovery process. This is the good news that a recovery is doable. Needless to say, PATIENCE IS REQUIRED.
So, let's begin. Recharge your device, uninterrupted, for at least an hour. (I let mine recharge overnight).
Make sure that your computer has the latest version of iTunes installed. If you're not sure of this, download and install the latest version straight from the Apple website.
After you've done this, restart your computer and wait for the desktop to return.
Connect your iPhone to your computer with a USB cable. I've found that the cable MUST connect to your computer's USB 3.0 (SS) port. (This procedure didn't work for me when I connected the USB cable to a USB 2.0 or 1.0 port. It had to be connected to a powered 3.0 SS port).
After you've done this, on your iPhone, physically hold down both the Power and Home buttons for a few seconds and wait for the white Apple logo to appear.
Once you see the logo, release your finger off the Power button, but keep holding down the Home button for a few moments until you notice that the iTunes program has appeared on your computer's desktop. This is proof that iTunes has finally recognized the existence of your iPhone that was stuck in Recovery Mode.
From there, iTunes will give you a pop-up message asking if you want to Restore and/or Update your device. Select to restore it.
Follow and answer the prompts.
Meanwhile, if you had previously saved your iPhone's profile or settings to your computer or to the iCloud, let iTunes continue to sync the saved history of your Apps through all of the steps it requires to go through.
Note: Don't let anything disturb the restoration process until iTunes tells you that it was successful. Don't fumble with the USB cable, don't mess around with your iPhone or the iTunes program while the restoration is in progress. Only give input when you are prompted to do so.
You'll know the restoration was successful when your iPhone feels very warm in your hand, temperature-wise. Press the Home button, swipe, and answer the prompts.
Hope this helps.
P.S. -- If this procedure STILL doesn't work, contact that certified Repair Tech, contact Apple Support (hopefully, your device's warranty is still in effect), or make an appointment with the Genius Bar at your nearest Apple Store.