"An exon is a segment of an interrupted gene that is represented in the mature RNA product."1
Exons are the segments of DNA that are copied onto RNA during translation, and thus are expressed ('expressed region'); or to put it more directly, exons are the segments of DNA that are made into genes. It would, in fact, not be too misleading to say that exons are genes.
On the DNA molecule genes are often 'interrupted' by bits of 'Junk DNA' that do not contribute to the final product of that gene. These interruptions are called 'introns'. Introns must be removed from the RNA transcript (which is a direct copy of the DNA) before it can be used to produce proteins. This process is called RNA splicing, and the final product is mRNA (sometimes, to be very precise, we say 'mature mRNA' or simply 'mature RNA'), which consists entirely of exons.
RNA splicing is a complex subject, but all exons have specific beginning markers, called start codons, and end markers, called stop codons. Introns also have specific markers at their ends2; once the RNA polymerase identifies the start of a gene, it simply removes every segment defined by these markers3 until it reaches the end of the gene.
A gene with no introns is called colinear; these genes are common in yeast and bacteria, but much less common in eukaryotes. We do not understand the reason that introns multiply in some genera and species so much more than others. However, it is worth noting that introns can collect mutations and/or multiply themselves at a much faster rate than exons, because there is no evolutionary pressure on them. This makes them useful for extrapolating genetic relationships among species.
1. Quote from Essential Genes, by Benjamin Lewin, Pearson Prentice Hall, 2006.
2. The most common markers for introns are guanine + thymine on the start or 'left side' and adenine + guanine on the end or 'right' side, so this is sometimes referred to as the GT-AG rule -- but the RNA uses uracil where DNA uses thymine, so sometimes the term 'GU-AG rule' is used. These combinations mark over 98% of splices, but there are other combinations.
3. i.e., After it recognizes a start codon indicating the start of the gene, it starts transcribing information; if it finds a GT segment before reaching the stop codon, that must be an intron, so that is removed. Once it reaches AG it starts collecting again... And so on, until the stop codon is reached.
This was written to help me understand some of the information about exons better. If you have questions or facts that you think I should add, please let me know.