The corpus callosum is a bundle of fibres that connects the two halves of the brain, the right half (which controls the LEFT side of the body and the left half which controls the RIGHT side of the body.
The corpus callosum is a communication pathway which allows the two halves of the brain to communicate because, as well as controlling different sides of the body, each half of the brain controls different functions, so the corpus callosum allows you to name what you see, for instance. For example, a picture or pattern is recognised by the right brain, but it is the left brain that allows you to name it as a robin or someone you know or a tree.
Quite often, you will not know that someone has had their corpus callosum cut because their eyes will tell both halves of the brain what they see, for instance. The right eye will tell the left brain what it sees and the left eye will tell the right brain what it sees. There may be problems with pattern matching though.
Some people are born without a corpus callosum and their level of intelligence is not necessarily affected, though many of these children MAY have retardation. The corpus callosum may also be cut in people with very severe epilepsy. This seems to stop the electrical scrambling of an epileptic fit from spreading throughout the brain.
There may be some effect on people who have had the corpus callosum cut but not necessarily always openly visible changes. There have been some experiments on rats but this kind of experimentation is not allowed on humans so the only knowledge comes from observations of those who have had this operation from necessity (so were already ill) or who were born without a CC and who therefore may have had other problems.