On a floppy disk, the boot sector (sometimes boot record) is the very first sector on the disk, sector 1, head 0, track 0. Generally it is 512 bytes long and must end with the two bytes 55 AA in order for the BIOS to load and execute it.
The BIOS (usually via interrupt 19h) will load this sector to 07C00h and jump to the that address in order to load the operating system. The boot sector program then locates and loads whatever is needed to continue the boot process.
The MS-DOS boot sector would load MSDOS.SYS and IO.SYS, Windows only loads IO.SYS, PC-DOS would load IBMBIO.SYS and IBMDOS.SYS, I believe.
For FAT filesystems, at least, the boot sector also contains the bios parameter block, which tells BIOS/DOS how to read the rest of the disk.
For hard drives, there is a master boot record here instead, which contains the partition table. There is a boot sector at the begining of each partition. I don't think there's anything stopping you from putting a MBR on a floppy, other than the fact that it would be terribly silly.
Back in the days when DOS reigned, the boot sector was a favorite spot for virii to lurk, where they could load themselves into memory before the operating system even got a chance. They would remain resident in memory, usually hooking an interrupt and infecting every disk put in the drive. Nowadays they'd have to survive the transition to protected mode, which is a tricky task indeed.
Thanks getha for correcting my head/sector/track number!