Answer:
Usually you can receive digital cable on a digital or analog (old) tv, but what matters is that you have a digital cable "box." The digital cable box will usually have its own remote that can be setup to control other things such as the TV or DVD player in addition to the digital cable channel and guide options.
And be sure the cable Company is sending you a High Def. signal to that box. Most companies will charge more money for the HD stations.