Hi Lisa, and Welcome to Tom's Hardware!
The quick answer is yes.
First tell us what kind of HDD is in your 'old' computer (Company, model, size, PATA or SATA) and if it was a boot drive (has an OS on it as well as data).
Then tell us what your present computer is (Company, model, what available internal ports you may have, and what external ports on the back as well as USB ports that are available).
Then one of us can give you clear instructions as to how to copy your old data over to either a new HDD, or USB flashdrive. And if you will need any additional "interface" device to connect the older HDD to the new computer.