How to know if my BIOS supports UEFI?

If you have Windows installed and usable, run the HWiNFO tool. Under the motherboard section (in the main window) it will say if the motherboard is UEFI capable or not (regardless of which mode the current Windows was installed or booted). If the HWiNFO utility displays 'UEFI', the motherboard is UEFI capable, otherwise it will display 'Legacy'. (Source)

Otherwise:

Put an Ubuntu 13.10 amd64 CD/DVD into the optical unit and on reset bring up the boot menu. If the PC has UEFI, the boot menu will present two items for the CD, one marked as "UEFI" and another "normal" one. If you select the normal one, you will get a graphical menu, while selecting the UEFI item will give you a textual menu. See this link for screenshots: https://askubuntu.com/questions/307508/how-to-tell-if-my-laptop-has-uefi/307930#307930

The other answers are incorrect. I have a UEFI PC with Windows 8.1, but no EFI partition, also the \Windows\Panther\setupact.log file has "Callback_BootEnvironmentDetect: Detected boot environment: BIOS".

Also, what if the HDD is inaccessible (it died and you replaced it with a new blank one)? Or you are assembling a new desktop PC and you haven't installed any OS yet?

There might be other tools to check the BIOS/UEFI type that might also work from USB sticks. I'll leave those to be an exercise for the reader...

There will be an extra UEFI partition on the hard drive, It may be hidden, if you can get a look at the files in the partition there will be some with an .efi file extension.

Different manufactures label the partitions differently, mine is labeled "HP_TOOLS"

I agree with William, all the new laptops I have seen in the last 2 years have UEFI

EDIT:

Another way may be to look in the C:\Windows\Boot folder for an EFI folder, it does not exist on non efi bios system's, You may have to unhide system or protected folders in folder options to see this folder.

I am having problems confirming this information, but, I have personally not seen a laptop released within the last few years (non atom and latest technology) that was not EFI.

The best bit of advice I can give is to write down the BIOS version/serial/manufacturer and try to Google it and see what you can find out.