You say you can boot with Cmd+R. Have you attempted anything with Disk Utility yet? If not, check out the paragraph below. Apologies for the slow reply!
If your Mac turns on, and the display works, but it won't boot, there could be many issues at play. But the one we like to rule out right away - or repair, if possible - is any problem afflicting the hard drive. The easiest first step on that front is to run Disk Utility. On a Mac running or later, you can run Disk Utility by booting into OS X Recovery Mode.
Make sure the Mac is off. (If it's not responsive because it's stuck on a grey, blue or white screen, just hold down the Mac's power button for several seconds until it gives up and shuts off.) Hold down the Command and R keys, and power the Mac back up again.
Eventually, you'll end up on a screen headlined OS X Utilities. (Once you see that screen, you can release the keys you were holding down.) Click on Disk Utility. Then click on your Mac's built-in hard drive in the left column of Disk Utility. (Usually, you'll see two listings for your built-in drive: the first includes the drive's size, like 500GB, in its name; and nested underneath it is your drive's friendlier name. You want that second one.) On the lower right of the Disk Utility window, click Verify Disk, and then wait while Disk Utility does its thing.
(Paragraph taken from Macworld)