Hi,
If your new computer doesn't accommodate the drive then you will need a three and a half inch hard drive enclosure to put the hard dive in from the other computer.
You will have to look at the drive to work out which type enclosure you will need.
It will either be a SATA connection like this >>>>
Or a IDE/ATA connection like this >>>>
If you compare the connections on these examples you will one has plastic connections and one has brass pin connections.
As soon as you know which type of drive you have you can then proceed to purchase an enclosure.
When you have got the correct enclosure all you have to do is install the drive and connect it your new computer.
You will be able to access the drive, copy your data and use the drive as an external hard drive.
Regards.