You need to access the setup screens on your router -- see literature which came with it or download manual.
Once in, go Wireless Setting , Wireless Security and read off the passphrase and note the security type and apply that to the new computer.
If the passphrase is starred-out suggest you create a new one and apply it to both computers.
If you have problems with the new computer connecting, try disabling wireless security in the router to narrow down the issues.