"Reformat" means that ALL the data on that old HDD will be lost permanently. So stop now and decide: do you want any of that information? Or, is it OK to destroy it all?
IF you are willing to destroy all the old data and use the HDD as if it were new and empty, it's not hard. Install the old HDD in your new machine normally. As you suspect, if it is a 2½" HDD from a laptop you will need a small adapter plate to fasten it into a desktop's 3½" drive space. But if it's a SATA HDD unit, the connectors should be the same and work OK.
Then you need to Install Windows fresh and any application software. Do NOT try to boot from that HDD. Put your Windows Install CD in your optical drive and set the BIOS to boot from that drive first, and the HDD unit second. When you boot into the Install routines, the first thing to do is to Delete any and all Partitions that exist on the old drive until it is all Unallocated Space. THEN you start the Install on this HDD. The process will Create a new Partition and Format it for you, then load Windows onto the drive..