Don't think of solar power as replacing all your needs, its one step towards reducing overall use of utility power. The solar array charges up the battery bank, this occurs throughout the day as the sun shines. Using new controllers, those things you have connected to the solar system can draw against the power being generated, as well from the battery bank. Once the sun stops, yes power creation stops, and all draw is from the battery bank. Your total usable power is a function of both how much you can create and how much you can store. Currently I am not a fan for solar to lead acid battery banks. The batteries are too costly and just do not last.
Other systems can be effective at reducing external power costs:
insulation of the home
use of high efficiency applicances
heat pump if appropriate
ground or water thermal pumps if appropriate
and this is cool:
http://www.wimp.com/energyleaf/