BIOS Whitelist Request
Some BIOSes have options within them which are "whitelisted" ie cannot be toggled. Some people want to toggle these options (eg fan speed,WWAN and wireless cards) so request the BIOS options to be "un-whitelisted".
Example:
From here
...Many manufacturers such as Lenovo and HP uses a list called "White-list" in their machines BIOS to limit what peripherals you can use.
From Beware the Dreaded White-List
So basically it's a list of "acceptable hardware" that may be installed in the computer. The most common internally installed piece of hardware is the network (Wifi) card, which is an optionally installed device that goes in an internal PCIe slot.
The thing is that if you swap the network card, upon booting the computer will check to see if that WiFi card is in the "white list". If it isn't, the computer WILL NOT BOOT.
BIOS Modding
3rd party modifications of the official BIOS, involving 'flashing' of the BIOS, which is very risky for the inexperienced May void your warranty. Failure in flashing the BIOS may "brick" your computer. Attempting to mod an RSA-signed BIOS (HP) will almost certainly cripple your computer.
Community example: bios-mods.com.
BIOS Unlocking Request
Request to enable to toggling of the whitelisted options in the BIOS. Some BIOSes don't even have the options whitelisted - they are just not displayed; hence people request so that a modded BIOS is made with the options added to the menu. For example, my Insyde H20 BIOS for my Acer laptop does not have an option to enable VT-X, so despite the CPU supporting VT-X I can't use it (though I got it to work eventually without using a modded BIOS).
Basically BIOS Whitelist Request is the same as BIOS Unlock, and to achieve that you use modded BIOS in place of the officlal manufacturer's BIOS.