The authorized is the amount that a company can issue. It isn't the amount of shares that are out there. The reason it is important to know the authorized is because if a company decides to raise it, and say they more than double what it is now, you can be sure that they plan on diluting and issuing more shares in the future. Especially in penny stocks, if you see the authorized being raised by a large amount be ready for some dilution.
The outstanding are the amount of shares issued. You probably have heard people touting that the os is low on a particular stock. The lower the outstanding the easier it is for it to move. If you have a stock with say billions of outstanding shares, there can be loads of volume and it will still not uptick. When the outstanding is low it will move more easy.
The float is just the outstanding minus the restricted shares. Restricted shares aren't being traded so it lowers the outstanding to a smaller amount thus making it even nicer if the float is small.
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