I sometimes have problems like this myself and it can be down to DNS.
Here's what to try and do!
On the machine that works do, click START > RUN > cmd
> OK > ipconfig
You will get a list of info "like" this (not identical):
Windows IP Configuration
Ethernet adapter LAN:
Connection-specific DNS Suffix . : xxx
IP Address. . . . . . . . . . . . : 10.0.10.62
Subnet Mask . . . . . . . . . . . : 255.255.254.0
Default Gateway . . . . . . . . . : 10.0.10.9
Note the default gateway address.
On the laptop that doesn't work do the same and the part that says IP address should be identical to the working machine apart from the last set of digits.
If not then your laptop is not picking up an IP address and may be set to static IP not DHCP (see DHCP settings below if this is the case)
If all is ok with the IP address then in the black window that you typed ipconfig in type this ping <insert default gateway address>
and you should get a reply similar to:
Pinging 10.0.10.9 with 32 bytes of data:
Reply from 10.0.10.9: bytes=32 time<1ms TTL=255
If you get a request timed out again see the DHCP Settings below.
If however you got a similar response, type this in to black screen ping www.google.com
and you should get a response similar to the one above.
If not then you have a DNS issue and what you need to do is find out the DNS address for whoever you ISP is (AOL, Tiscali etc) you can do this by ringing them or searching google for your ISP name followed by DNS address.
To use these addresses see below for DHCP settings.DHCP Settings
To set you machine to use DHCP instead of static right click on My Network Places, select properties, then right click on your wireless connection and again select properties.
In the screen that comes up find the line in the white box that says "Internet Protocol TCP/IP" and double click on it.
Now check the box that says "Obtain an IP address automatically"
If you were having DNS issues then towards the bottom of the screen select "Use the following DNS server addresses" and input the DNS IP addresses you got from your ISP.
I hope this helps and happy surfing.