I don't know how familiar you are with wireless networking, but have you actually "Connected" to the wireless network?
In other words, have you run the wireless adapter's software, scanned for "Available Wireless Networks", found and highlighted the one that matches your router's SSID (network name) and selected the option to "Connect" to it? You should be prompted to enter any WEP or WPA encryption key and, once "Connected", the wireless adapter should get its IP address by DHCP from the router and you should have network and internet access.
There is a difference between the Wireless Network Adapter and Router communicating with a good signal and "Connecting" (as in "logging in") to your Wireless network.
<<< Also, does anyone know what it means on this laptop if the wireless light flashes? >>>
It could just be an indication that data is being transferred, i.e. there is activity.
There are many reasons for dropped connections and often interference is a problem. Make sure you don't have a mobile or cordless phone located near the router or the laptop. You could try changing the Channel number used by the router - the default is normally "11" so try something like "7" or "4".