Sorry if I missed this someplace, but what is the model number of the computer itself?
If you can go to the vendor's site for the model you can determine what the original configuration was. I don't know if you have both integrated graphics and an installed card -- since you seem to have both Nvidia and ATI files .
The card itself is hardware, it is an installed card on the motherboard.
You won't be able to install drivers since it is not being detected. That you have graphics at all suggests there may be an integrated graphics chip as well as an installed card
Removing and reseating graphiics cards can be tricky -- and it depends on the type of card.
This is why you really need someone with hands-on hardware experience.
Typically you follow normal instructions for shutting down, unplugging and opening the case. This last can also be easy or difficult depending on the vendor. Dell's are usually pretty easy and the manual shows how.
Once you have identified the card in the PCI slot (most will have small fans on them and you will see the cable to your monitor attached to the card's external connections) -- there may be a clip or tab near the right or left bottom rear of the card connected to the motherboard -- this needs to be released as well as the normal PCI clip which holds cards in place.