The Airport base station is Apple's wireless router. An Airport Extreme card is the wireless card inside your computer that lets your computer connect to the base station (or any wireless router), wirelessly, and let you surf the net (if you have it set up to your router), and share files over the network itself.
If your current ADSL Modem is also a wireless router, you have no need for an Airport Base-station. The Net Gear will work fine.
Your speed will not be faster than the speed of your ADSL connection, but it may vary depending on if your current ADSL Modem/Wireless router is 802.11B, G or N compliant. The airport card in your Mac should be at least 802.11 B & G compliant and may even be N compliant as well. The airport card in your Mac will conform to whatever the Router supports and runs at, so if your router is set for G then your airport card in the Mac will work at G speeds and so on.
No, an Apple router won't be any faster than any other (well any other that supports 802.11n... but I suspect you won't be able to use 802.11n on your PB anyway).
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