BIOS for PSX is the same thing as BIOS for your PC.
It allows machine startup, displays boot screen, and provides services for applications to access the hardware. It's a piece of software that allows other applications to start and operate.

PCSX and PSX4All have something called HLE built-in. It's a reimplementation of original PSX BIOS in emulator software. It's mostly complete and allows running most games. But some games using bizzare BIOS features may need a real BIOS from a real PSX. This is when you need to provide the scph1001.bin file.


As for the so called "ROM files":
PSX does not use ROM files. Other game consoles used hardware cartdridges with some ROM variant chip inside storing the game code and data. Dumps of content of this chip are called "ROM files".
PSX does not use cartdridges but compact disks. So a dump of a CD is your familiar CD image file you usually use to burn CD-Rs. PCSX and PSX4All are able to use these image files directly, without burning to CD. You may use most image files format different imaging software use. Just pick your favorite one and dump your PSX CD to file.