Getting the Android-x86 emulator networked properly from within VirtualBox has proven to be more of a challenge than I'd anticipated. Fortunately, I've figured out exactly which steps need to be taken in order to successfully get Android-x86 to connect to the Internet from within VirtualBox...

1. Make sure the computer that is running VirtualBox has a stable connection to the Internet.
2. Open VirtualBox, but do not start Android-x86 yet.
3. Select your Android-x86 virtual machine and click Settings.
4. Select Networking and ensure Attached to: is set to NAT.
5. Click Advanced and make sure your Adapter Type is set to PCnet-FAST III (Am79C973). Mine was set to Intel by default, and this is what kept me from getting connected for so long.
   
6. Click OK.
7. Start your Android-x86 emulator.
8. Once booted up, hit Alt + F1 to open the Android console.
9. Type ifconfig eth0 up and hit enter.
10. Type dhcpcd -LK -d eth0 and hit enter.
11. Type setprop net.dns1 [DNS IP] and hit enter. Replace [DNS IP] with the IP address of your DNS server. This is typically your router's IP address. So, for an average home Linksys router you would type setprop net.dns1 192.168.1.1 and hit enter.
12. Press Alt + F7 to return to the Android emulator.
13. Browse the Internet!

Obviously, an Android emulator isn't worth much if it can't connect to the Internet unless you're a developer. This process of tweaking the VirtualBox network settings got the Internet working for me on Android-x86, and hopefully it will work for you too.