A full metal jacketed bullet is formed from the "cup and core" method. The cup, or jacket, has a soft lead core swaged under high pressure into the jacket, normally from the base of the bullet. The mouth of the jacket is normally crimped over the exposed lead core slightly to hold everything in place. Lately, some companies have been installing a copper disc over the exposed lead at the base of FMJ bullets, totally encapsulating the lead core to reduce airborne lead exposure.
A Total Metal Jacket is a soft lead core that has been swaged into the final shape and size under pressure. It's then electroplated with copper, which completely covers the lead. The electroplating is just a few thousandths thick, and the jacket isn't as heavy as a jacket used in the cup and core method.
This the reason TMJ bullets can't be driven to the same higher velocities as FMJ bullets. You're usually restricted to about 1,250 fps with plated bullets in handguns. They will strip through the rifling if driven much faster and tumble once they leave the barrel. I've had plated bullets tumble at 1,450 fps from a 357 Sig, but I'm able to run jacketed bullets at 1,533 fps through my 9x25 Dillon pistol. You also can't crimp TMJ bullets at the case mouth as heavily as you can a FMJ bullet. If the crimp is too severe, it will cut through the plating and the core will separate from the thin jacket when the bullet leaves the muzzle, causing an inaccurate bullet.
Neither TMJ or FMJ bullets are designed to expand on impact. They are intended to function in all pistols, and are best used for targets and plinking. The exception would be when used against signers of the Geneva Convention during time of war, where FMJ bullets are pretty much the only game in town, but there is also some question about that rule, too.
Hope this helps.