A deep-think is something that happens when a sim experiences a physical interaction that takes a long time for it to compute. This will affect sim performance negatively, causing things to "slow down" or "lag" within the sim (especially physical things such as avatar movement). It can happen when many physical objects are interacting at the same time, when a physical object is stuck or wedged somewhere, or when a complex shape (such as a torus) is interacting weirdly.
If Second Life ever upgrades to a more recent version of Havok, deep-thinks may be eliminated.