Intelligence is defined as the ability of a mind to reason, or to acquire and apply knowledge.
It is also used to describe incorporeal minds, especially in the Intelligent Design debate, which postulates that an "Intelligence" must have created the universe. Therefore it can mean God, Allah, Saint Peter, Vishnu, or the Flying Spaghetti Monster, depending upon which religion you subscribe to.
Intelligence as part of a brain
Biologically speaking, intelligence is that property that results when neurons get together to form a complex brain. The neurons act together to form patterns that represent information. Other parts of the brain can recall and change that information, or put it to use.
The intelligent brain is also good at reasoning. In other words, it can detect patterns between seemingly unconnected pieces of information, form links, and theorise about them.
The extremely intelligent brain also exhibits consciousness, which can be loosely defined as knowledge of oneself, and the desire to ask the question "Who am I?". So far as science can tell, Homo Sapiens is the only creature that asks this question and seeks to answer it for himself. Some seek the answer in religion, some in science, some in philosophy.