A way to look at support and resistance levels is in terms of supply and demand. As the price of a stock falls, the price becomes more attractive. At some point, investors start to feel the price represents fair value again. At any price below that, they start buying. That price can be thought of as the support level. At fair value the supply and demand have come into equilibrium and the price should find support, that is it should stop falling.
Resistance, as you will have guessed is the opposite of support. As the price of a stock rises, it becomes less attractive. The higher it rises, the less attractive it becomes until it reaches a price where there are no more buyers. This is the resistance level.
Support and Resistance levels represent reversal points. When support levels are hit, the price starts rising again. At resistance levels, the price of the stock starts falling again. So you can see why technical traders would buy at support levels and sell at resistance levels.
So what happens if the falling price keeps falling through the support level? It usually means that investors have new information that leads them value the security at lower levels. Eventually a new support level will be found that reflects the new fair value for the stock. In cases of bankruptcy, that level may be zero.
Sometimes the stock price breaks through the resistance level as well. When that happens, the resistance level becomes the new support level. As above, it usually is a reflection of new information in the market that raises investors' expectation for the stock or tradable security.
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