Question: How does a BIOS get updated? Is it by microprogramming or microcoding? If not, then what happens?

Answer: 
On a modern Intel platform, the BIOS is stored in a SPI (Serial Peripheral Interface) NOR Flash memory chip attached to the PCH. The exact method of update varies from BIOS vendor to vendor, but I used to maintain an AMI UEFI-based BIOS for my company. The firmware update tool (AmiFlash) could be executed from Windows, UEFI shell, or a self-booting DOS environment and simply overwrites the content of that SPI flash.

In the past, Intel platforms used to have a FWH (Firmware Hub) based storage device, which sat on the LPC (Low Pin Count) bus.

Microcode updates are often delivered as part of a BIOS update, and are applied as binary blobs on each power cycle -- the actual CPU is not modified in any way, shape or form.

I imagine AMD is quite similar -- they have probably moved to conventional SPI memory as well.

If you take a power-cycle or other interrupt in the middle of an update, you'll brick your platform. Some motherboards have twin SPI flashes, allowing an A/B bank-switching style approach to be used.
Firmware updates are essentially the flashing of a chunk of memory onto a flash memory chip, or EEPROM device on the mother board. The bios code is very critical, as it's the startup code to get the computer up and running for the specific OS (windows, Linux, etc). Also the bios handles many other functions, critical I/O assignments, boot drive selection, etc.

The operating system has an application that handles the actual update of the BIOS code, you are able to download the new code (in hex format) from a web page, or possibly via a USB thumb drive, and the mother board manufacturer will provide a program that talks to it's hardware, and 'pushes' the new BIOS to the mother board. 

Many high end mother board BIOS update routines have a failsafe method of updating, in case the device loses power during the update. The new code is written to a empty area, then a check sum is downloaded. The new code goes through the check sum calculation to verify it's correct. If it is correct, a flag in the bios points to the new area of memory. Then the computer is rebooted, and the new code is used. If anything goes wrong, the flag will still point to the old code, and the user is informed to re-apply the update. 

I wrote and maintained update code for embedded 'black-boxes' used in commercial transportation, medical devices, and multi-media devices; mp3 players, set top boxes, etc.