I'll post the next step now and we can catch up tomorrow then.
ping 127.0.0.1 (sent 4, received 4, lost 0)
What you did here was ping the loopback address of the adapter. Hint: A lost = 0 value is ideal. Since this test was successful, it is telling us that Vista has properly installed the TCP/IP stack on it. This is goodness.
ping 10.0.1.3 (sent 4, received 4, lost 0)
Being successful, this is confirming that the adapter's device driver is not corrupt or missing. Also goodness.
ping 10.0.1.1 (sent 4, received 3, lost 1)
Being mostly successful, this is telling us two things: 1) The adapter can find and communicate with the router, and 2) Because there was at least one packet lost, your wireless network may be affected by some type of Wi-Fi interference. Being that I assume that the laptop is near the base station, I would have expected 0 packet losses.
ping 66.102.7.104 (sent 4, received 2, lost 2)
Still somewhat successful, this is telling us two things: 1) The adapter can find and communicate with the Google web site by using the website's IP address. This would also mean if you fired up the HP's web browser, you would have opened the Google page by typing in the IP address, but not the normal method of entering: www.google.com (i.e., it's URL), and 2) With half the packets being lost, the connection between the router to the Google site is somewhat "dirty."
ping www.google.com (Ping request could not find host www.google.com. Please check the name and try again)
100% of the time when this ping fails, it's because the DNS IP address information is either missing or corrupted. Think of DNS as the Internet's version of the Yellow Pages. As you saw in the previous ping command, you can access websites by their IP address, but not their URL when not using DNS.
Ok, I don't know why (possibly because of sensitivity to Wi-Fi interefernce) the adapter is not getting the DNS IP address information from the AirPort Express' DHCP Server service, but you do have a workaround that should get you by.
... and that is to enter the DNS IP address information manually instead of it being provided by DHCP. To do that on a Vista computer, just do the following:
o Right-click on the Network icon in the System Tray.
o Select the Network and Sharing Center option.
o Select the Manage network connections option.
o Right-click on the appropriate connection, and then, select Properties.
o On the Networking tab, click on the Internet Protocol Version 4 (TCP/IPv4) option, and then, click on Properties
o On the lower half of the Properties page, select the radio button option "Use the following DNS servers addresses to enable it," and then, enter the following IP addresses: for Preferred DNS Server, enter: 208.67.222.222, and for Alternate DNS Server, enter: 208.67.220.220
o Click OK, and then, click OK again to close the various windows until you get back to the desktop.
You may need to restart the computer for the settings to take place. Please do so, and try again to access the Internet via a web browser by entering a normal URL instead of an IP address. Were you successful?