One-piece toilet. Two-piece toilet
There is virtually no difference in the performance of one type over the other. The two-piece toilet has been around "forever" but the one-piece toilet is gaining ground due to it's more modern look and, in most cases, lower profile.

CLASSIC TWO PIECE TOILETS
Two-piece toilets have, as the name already suggest, two parts, the lower bowl and upper toilet tank that are bolted together with two and sometimes three bolts. Bolting the two parts together should be done carefully. If they are not fastened tight enough the toilet tank will leak. If fastened too tight the tank will crack and you will end up buying another toilet.
The area where bowl and tank meet is hard to reach for proper cleaning and/or disinfecting.

CONTEMPORARY ONE PIECE TOILETS
As a rule, one piece toilets have a lower profile than two piece toilets. The one piece toilet is designed so the tank and bowl are cast as a single porcelain fixture. One piece models have a single, integrated tank and bowl design for a more contemporary look. These units are more expensive than the two piece toilets. They are also heavier than the two-piece toilets which makes it harder to install. Because of its design there are no hard to reach areas for cleaning.

NOTE
Wax ring, "T" bolts (also known as closet bolts) and toilet seat do not come with the toilet and must be purchased separately.