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BIOS is responsible for initialization and startup of the module, and maybe the boot process, and it provides some basic firmware interfaces to interact with the hardware, like the old D*S software interrupts, but the BSP is the layer between the OS and the hardware, which translates generic OS functions to special hardware commands, which in turn might be handled by the BIOS...
BSP comes close to a hardware abstraction layer thing...
* Firmware is equivalent to "software" in PC world. Unlike PC's, user can't load whatever he/she wants on the embedded board. So software on an embedded system is "firm", this is why software called as firmware in embedded world.
* BSP -board support package- is a bunch of patches/modifications to adopt a sw system (usually O.S. like winCE, linux) to your custom board. i.e. you have a board with T.I.'s Omap processor, linux kernel already have Omap support, but you need to tailor it for your board. You have different display, different keypad etc. All these changes constitues the BSP.
* BIOS basic input output system, orginally a PC thing. It's the first piece of program runs when the computer powered up and responsible for basic HW configuration and bootloading. I've heard some terms like DSP BIOS, but normally BIOS means just PC BIOS to me.