Do you recall which update you were installing and which version of Mountain Lion you are running (10.8.?)? Can you jot down and report the information on the screen?
Do you have a reasonably current backup (Time Machine or other)? If you don’t have a backup you should make creating one a high priority, especially since your machine is acting up. This is covered in “Backups” below.
If you get stuck in any of these steps or don’t feel like proceeding, then post back what happened and someone is likely to help you take the next steps. Also, at any time you can stop and take your Mac to an Apple Store where they will diagnose the problem for free, fix it for free if it does not need hardware repairs, and if it is the hardware they will diagnose it for free and give you and estimate for repairs. But there are some very easy steps to take to get started.
Start by forcing a shutdown by holding the power button down until it shuts off. Then try rebooting and see what happens.
There are multiple trajectories from there depending on what happens. It could boot normally. It might boot in Safe Mode (explained below). It might hang on a gray or white screen. There are different steps you can take depending on these results.
No matter what the problem, there are five simple and safe things you can try before you need a deeper diagnosis.
If it does not boot normally there are five universal cures that do not hurt and might help:
If the system is not off, force it off by holding the power key down until it turns off.
1) Reset SMC: http://support.apple.com/kb/HT3964
2) Reset PRAM: http://support.apple.com/kb/PH4405
3) Run hardware diagnostics: http://support.apple.com/kb/ht1509
4) Check and repair the system disk:
Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.
When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported then click on the Repair Permissions button. When the process is completed, then quit DU and return to the main menu. Select Restart from the Apple menu.
5) Boot in Safe Mode: http://support.apple.com/kb/HT1455
After taking these steps try booting it normally. If it works, then that is very good news. Do not install the update until you have a current backup and the disk check/repair finished normally (without reporting errors). Jump to “Backups.”
It might boot in Safe Mode but not normally. If so boot it in Safe Mode and jump to “Backups.” Post back for further instructions.
If during a boot it shows a white screen with a question mark, a folder blinking, or a gray screen then post back and wait for further instructions.
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*** Backups ***
If it will boot in normal or Safe Mode then update you current Time Machine backup by plugging in your Time Machine external disk and running a backup.
If it will only boot in Safe Mode and you have a wireless Time Capsule, you will not be able to access it in Safe Mode and will need to wait until you can boot normally to reach it. If this is the case stop, report back and wait for further instructions.
If you do not have a Time Machine backup or a bootable clone disk it is time to create one. You will need an external hard disk you are willing to erase an dedicate to be your Time Machine backup disk.
See http://support.apple.com/kb/ht1427 and https://discussions.apple.com/docs/DOC-4055
For more about backups
See https://discussions.apple.com/docs/DOC-3045 ,
and https://discussions.apple.com/docs/DOC-6031
and Methodology to protect your data. Backups vs. Archives. Long-term data protection