After a thunderstorm, ice storm or high wind, the most urgent calls are trees or large limbs resting on a roof, a car or a fence, and trees split down the trunk that could fall at any moment. If a tree is touching or tangled in power lines, stay well away and call the electric utility first. Most tree companies won't work near live lines until the utility has made them safe.
For everything else, an emergency crew can make the situation safe, take the weight off the structure and tarp or clear what's needed, then come back for the full cleanup. Take photos of the damage before anything is moved, since your homeowner's insurance may cover removal of a tree that fell on the house. Expect emergency and after-hours rates to be higher than scheduled work, and ask for the price before the crew starts.