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Ohms is a measure of resistance. You don't think of it, but substances conduct or resist electricity on a scale. Rubber isn't very conducive, copper is. This is why some premium cable are made out of silver and such - these substances conduct electricity better than others.
When the electricity passes into the speaker, some of it is 'resisted.' The ohms rating of the speaker is how much is resisted, and an indication of how much energy it takes to drive it - the higher the ohms rating, the more difficult it is to drive.
An amplifier amplifies the electricial signal, causing the speaker to vibrate with greater force, otherwise the impulse would be barely audible. How much the amp drives is measured in watts. When an amp manufacturer gives it's wattage rating, it's always at a specific Ohm resistance. 100 watts @ 4 ohms, for example. When the resistance is 4 ohms, this amp will produce 100 watts. If that resistance changes, the amp will produce a different amount of watts.
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