This may be due to security settings - whilst the Kindle will happily connect immediately to unsecured wifi, you need to tweak the settings for secured connections (which I'd hope you are using
In the Kindle, you can add a new wifi connection manually:
Go to Wifi settings, and if your current wifi shows up with the option to forget, cursor down to it and select "forget".
Now go down to "Enter other wifi network"
In the box that pops up, put the SSID of your wifi (this may be set to the router's default, or you may have changed it)
Leave the password blank for now.
Cursor to "Advanced"
In this box, the connection type should be DHCP (which will blank out the next 4 lines).
Change "Security type" to match the security used by your router (WEP, WPA or WPA2)
Put the password in and connect.
If you let us know which router you're using, we can probably explain this in more detail.