A depression is a severe sustained recession. There is no clear dividing line between a recession and a depression, the term depression is used for a recession that is sufficiently deep and goes on for long enough.
A depression is severe, and sustained, but everything that occurs in a depression also occurs in a recession to a lesser degree: a shrinking economy, a rise in unemployment, etc.
In general depression will last for several years (from the beginning of the contraction to the recovery to pre-depression levels), and problems that would be rare in a mere recession are likely to be widespread: failures of banks or large businesses (actual insolvency, rather than just a few years of losses), high levels of government debt or financial crisis at the national level, or similar large scale problems that take years to solve.