What part of the brain is responsible for processing information about bodily sensations?

The answer to this question would be somatosensory cortex.

In somatosensory cortex, the brain is processing the signal from sensory nerve cells from the body. This input mostly coming from the skin. The area dedicated is not affected by the size of the organ. The area of the skin with more sensitivity, like the hand, will have much more area in the brain.