How does the CMOS jumper work?

In a desktop PC the chip that BIOS that stores the settings requires an external power source in order for it to retain whats stored on it. The clear CMOS jumper is there just in case you can't actually get into your BIOS to reset it though the menu options. Moving the jumper disconnects the battery and I believe it also shorts a pin on the chip to ground so you don't have to wait for the residual voltage to drain off. If you just remove the battery you need to wait 5 or 10 seconds for the settings to be lost/reset. With the jumper its instantaneous. Just make sure your power supply is turned off and unplugged before using it. And don't forget to put it back in the original position before you power up or you could damage your motherboard.
In a laptop its a little different. They use an eeprom to store info like the BIOS password. It won't lose those settings if you remove the battery. Thats done so you can't easily bypass/reset the BIOS password. Something you will be thankfully for if your laptop is ever lost or stolen. An eeprom must be powered up to be reprogrammed.

I beg to differ over your definitions. An Eprom does not need an external power source to retain it's data. It has a quartz window that allows it to be erased by UV light and normally has this covered by a label to prevent the UV content of ambient light gradually corrupting the data. The Eprom contains the BIOS default settings.

It's the CMOS memory that needs the power source (usually a couple of Lithium coin cells) to retain it's data. The CMOS holds the BIOS settings set by the user and when you perform a reset to defaults the data in the Eprom is copied to the CMOS.

EEprom is, as you say, electrically eraseable and reprogrammable. It also doesn't need an external power source to retain it's data. It's often referred to as Flash Memory.