The first is legacy BIOS: the BIOS will load the first 512 bytes of the boot sector, called the Master Boot Record (MBR) into memory location 0x7C00; they are then responsible for bootstrapping the rest of the system. Usually by doing the bare minimum to load a second-stage bootloader off the disc, and letting the second-stage bootloader do the rest of the work, since 512 bytes (actually only ~440) is really not very much.
The second is UEFI. The firmware will load a PE-formatted executable from a .efi-suffixed file on a FAT32 partition which has the EFI System Partition (ESP) flag set. Since this PE file can be any size, a second-stage bootloader is not generally necessary (but may be interesting for aesthetic reasons e.g. if you don't like PE).