It would depend. Does your camera have a removable media card (SD, Compact Flash, etc.). If so all you need to do is plug the card into your computer, assuming you have a card reader), you can also buy a card reader that plugs into a standard USB port. Make sure that if you go that route to buy one that reads your particular media card.
If your camera does not have a removable card you likely were using a card to connect to your computer via either a USB port or possible a serial port for an older camera.
In that case you will need the software and drivers to communicate between your camera and the computer.
Not all is lost. Many times you can go to the camera makers website and download the software and drivers to reinstall onto your computer.
What is the make and model of the camera?