|Based on your description I would say that your new drive has a SATA interface and your PC has IDE, or at least the drive you are replacing has IDE.|
If you want to keep that drive, and there aren't any SATA connectors on the motherboard, then yes, you would have to get an adapter, or a SATA addin card if you have a vacant slot.
Also, the power connector is most likely different than the old drive has, so you would have to get an adapter for that.
Easiest thing to do is to return the new drive if you can and get another drive with an IDE interface.
...if it ain't broke, then you ain't having fun...