When the computer is turned on, the BIOS (which is "just" another program, which is the first one run) is read into memory and executed by the CPU. (I have no idea how this is done, sorry.) It then performs any preparations or tests needed, according to the settings which are set up.
Those settings (which are incorrectly sometimes called "the BIOS") are what you can control when you boot your computer, by pressing e.g. Del or F10. They specify which devices the BIOS allows booting from.
Upon reading the settings, the BIOS tries, in order, to read the first sector of each device (called the Master Boot Record on most hard disks, or just "sector 0"), checks to see that it's bootable (it must end in the hexadecimal number 0xAA55), and if so, it just copies the sector to memory and transfers the control to the first byte of the data. After that, the MBR is on its own, and needs to load whatever that is needed.
Often times, for hard disks, this means that the MBR must parse the partition table and then read the first sector of the active partition to which it must boot, and then transfer control to that sector (called the boot sector). The boot sector then reads any files needed from the partition and calls the boot loader, which loads the operating system.
Of course, this last piece is not required. The device's boot sector can just do whatever it wants, and in the case of CDs or DVDs, other standards are in place that govern what is happening, e.g. El Torito. Different devices behave differently after they get control from the BIOS, so it's hard to say what happens after that without knowing about a specific type of device.