Not necessarily if the cable is meant to be used between modem and router. That cable would be a crossover configuration
Not so. A crossover cable is only required when connecting like devices. Which is to say, if you're connecting one PC directly to another (ie: NIC to NIC) or one switch to another. Although in the case of modern switches which are "auto MDIX" capable, that's not absolutely necessary, although I always recommend it.
A modem and a router are two different devices and as such, a regular (straight through) patch cable is all that's required.
For the Original Poster and sumgai:
First, do as Jennifer said and ascertain if your NIC is actually working correctly. Check in Device Manager.
Confirm you have the correct cable (compare both ends are the wires exactly the same - if so, it's a "straight through, patch cable"). If your NIC appears to be working properly and you have the correct patch cable and it's still showing the NIC as not being connected, try another cable. Preferably one you know is working.
If you suspect a driver issue, go to the manufacturer's website and download the most recent version and install it.