How can I get Google's two-step verification to work? On my Android phone, I can't seem to use my Google Account because I have the 2-step verification enabled. Can I log into Google without disabling the 2-step verification system?

Here is a link to Google's 2-step verification system:
http://googleblog.blogspot.com/2011/02/advanced-sign-in-security-for-your.html

Logging into my Google account is not a problem using the 2-step verification on a standard PC/Laptop, i.e. on a Microsoft/Mac OS, I just can't seem to use the 2-step verification system  on Android 2.2.2.
The answer is that you have to use a standard computer, log into your google account 2-step verified google account, go to authorise applications and sites, and set up your android device.  Google have said that 2-step verification will not work yet on android, so they have provided this work around for now.

Here is the link where you create a password to authorize your device:
https://accounts.google.com/IssuedAuthSubTokens

They will then give you a password which you can enter into your phone, allowing you to keep 2-step verification enabled.