Bottom line
The card you carry depends on your status. On active duty or in the SELRES, you get a Common Access Card (CAC). Retirees and dependents do not get a CAC. They get a DD Form card instead. Which DD Form depends on whether you are a retiree drawing pay, a retired reservist waiting on pay at 60, or a dependent. The card is government property, so keep it on you and turn it in if military authority requires it.
What it means for you
If you drill, you carry a CAC. It is your ID and your key into buildings, controlled spaces, and DoD computer networks. If you are in the IRR, expect no CAC until you get orders for AD longer than 30 days. When you retire, your card switches to a DD Form, and the exact form depends on your retirement situation. Your dependents get their own DD Form card.