Most lenses aren't perfectly sealed which means you can get moisture on the internal glass elements as well as the front and this can take a while to clear, leading to mould forming which is very bad. You're unlikely to damage a lens if this happens occaisonally but it's just better to avoid condensation in the first place.
Condensation occurs when moving from a cold area into an area with warm moist air as the cold lens elements cool the air causing moisture to precipitate out. As stated previously, good practice is to put the camera and lens in a sealed plastic bag before moving from the cold area. This traps dry air around the lens and gives the glass time to warm up. Do not seal the camera in a plastic bag when moving from a warm area into the cold! As the warm moist air inside the bag will start to condense as the camera cools.
Just because warm air can hold more moisture doesn't mean it will always be more humid, so there will be times when this is uncessesary, however it never hurts to be cautious. Also note that if you are in an extremely cold climate it can take a long time (hours) for all the camera internals to heat up again.
In general rapid extreme temperature changes are bad for equipment as they cause expansion and contraction of materials and can cause brittle components to break. For this reason I generally leave gear in a bag when moving around if possible as this will slow down heat transfers to safe levels. Also black camera bodies can heat up very quickly if left in direct sunlight (this is supposedly why the Canon super teles are white) so that's another reason to use a good bag.
Batteries can suffer in cold climates however these affects are usually reversible when the temperature goes back up. It's a good idea to keep batteries in an inside pocket when shooting in the cold.