This is probably a result of a corrupted filesystem -- the OS tries to repair it in order to start up, fails, and shuts down as a result. This in turn was probably caused by force-restarting the computer.
You can check this by starting in Recovery mode with Command-R, then running Disk Utility, selecting the startup volume in the sidebar (note that the volume will be listed under the disk it's on) -> First Aid tab -> Repair Disk button; this runs essentially the same repair process (and therefore will probably fail), but it'll tell you what it finds.
Assuming I'm right about the filesystem corruption, the next question is whether you need to recover any data from the disk. Is there anything on the computer you care about? Is it backed up? Are you sure it's backed up? If you need to get anything off the computer, do this before doing anything else! Getting data off an unbootable computer tends to be tricky, and depends greatly on what resources you have available (e.g. other Macs, external disks, etc).
Once you have everything you need copied off the computer, go back into Recovery mode, run Disk Utility, select the startup volume -> Erase tab -> set Format to "Mac OS Extended (Journaled)" -> Erase button. When that finishes, quit Disk Utility, select Reinstall Mac OS X, and follow the instructions. Note that this will download the installer files (about 4GB) over the internet, so it may take a while.
Relevant Apple KB articles: About OS X Recovery, Resolve startup issues and perform disk maintenance with Disk Utility and fsck, Erase and reinstall Mac OS X
EDIT: data recovery in a situation like this can be rather messy, with a number of things that need to be dealt with.
- Is the external disk in Mac OS Extended format? If not, you may need to reformat it to get full compatibility with Mac files. This is especially if you're going to use it as a boot disk (see below); in this case you'll also have to make sure it's formatted with a GPT partition scheme. This will, of course, erase the current contents of the disk, so if there's anything important on that, it needs to be backed up first. And be very very sure you're erasing & reformatting the external, not internal, disk.
- You need to get the internal disk to mount, so you can get at the files on it. OS X will sometimes refuse to even mount a badly damaged disk. If it's not mounting at all, my go-to tool is Alsoft's DiskWarrior, which is good at both repairing the file structure in place, and also (if it's beyond that) mounting a virtual cleaned-up version of the volume that you can copy files off of.
Finally, you need some way to actually copy the files:
- If you're familiar with the command line, you can use Terminal in Recovery mode for this; but if you're not already familiar with the command line, this is not a good time to try it out.
- You can also use Finder, but it's not available in Recovery mode. What you'll probably have to do is use Recovery mode's "Reinstall Mac OS X" option to install OS X on the external disk (after making sure it's in GPT+Mac OS Extended format), then reboot with the Option key held down & select the external as your startup disk. Note that you could also download, install, and run DiskWarrior from the external if necessary.
- You can use Migration Assistant to copy off your entire user account. It can also copy things like applications; but I'd recommend reinstalling them clean rather than copying possibly-damaged ones. As with the Finder, this isn't available in Recovery mode, so you'd need to install OS X on the external. Once that's done, you can either run Migration Assistant during the initial setup of the new system (one of the screens it goes through is labelled something like "Transfer information from another computer?"), or set the new disk up with a different account (choose a different account name to avoid conflicts), then run Migration Assistant from the Utilities folder.