It is called a slave drive. Is it a SATA drive? Hopefully you'll be using it for storage and not as your boot-device.
If it's SATA, you can just plug it in to a SATA port and a power connector and it should auto-detect the master/slave status of both drives. Assuming your newer drive is in port-0, that should automatically become your main drive, and any other port would be considered a slave. Boot to Windows and it will install the driver, if needed, and you should be able to access it as your D: drive (or whatever it assigns).
If it's PATA (the wide flat plugin with 40 pins), then there's a little more involvement, but I'm going to assume that your drive is SATA. If it's not, then just say so and someone will give you instructions
"Master" and "Slave" are terms that mean something ONLY on IDE ports. They are the labels we use to identify uniquely the two devices that share one IDE port and cable. These are set by jumpers you place on pins on the back of an IDE device. EACH IDE port MUST have one Master device if it is being used at all, and MAY have also a Slave device.
There is no such thing as Master or Slave for SATA devices. A SATA port can only have ONE device attached to it. There is no such thing as one Master drive for the entire computer, with additional Slave drives.
Your computer DOES have one unique drive in it - the boot device where the OS is stored. This does NOT make it the machine's Master! It may be called the Boot Drive. (Well, some people actually have more than one boot drive with different OS's installed on each and they choose which one to boot from as they boot. But that's more complicated.) All the other drives in the system usually are just data storage devices and all are equivalent in that respect - BUT they are NOT Slaves!
Most SATA drives now do have some pins on their back edges, some including jumpers on them. These are NEVER used for Master and Slave settings the way IDE drives work, because there is no such setting. In fact, those pins are used for very different functions on SATA drives, and changing them can cause you real trouble if you don't know what you're doing. For example, on one maker's SATA units, a jumper on a particular pin pair puts it into a mode reserved for RAID arrays in servers with special controller cards, and the drive simply will not turn on until it receives a special command from that controller card. To a "regular" SATA mobo port it will appear to be a dead drive!
So, to answer your main question, YES you certainly can put your old HDD into your new machine and use it normally. How depends on what type of drive it is, and what mobo ports you have available.
IF your old drive is IDE and IF your mobo has an IDE port on it, check out my post here from Sept 25/11:
It is addressed to someone who MAY have an optical drive already on an IDE port, but it covers what you need to set up an IDE drive. You will also need to connect to the drive a power supply from a 4-pin Molex connector from the PSU.
IF, on the other hand, your old drive is SATA, You can ignore all the discussions of setting jumpers and simply connect a SATA data cable from the drive to a mobo port, and a SATA power supply cable from the PSU. IF your mobo has more than one SATA port controller there MIGHT be a thing to watch out for. Most such mobos have one group of ports with labels like SAT0 to SATA3, or SATA1 to SATA4, that are controlled by the main chip and always work. Use one of those ports if possible. There may also be other SATA ports with higher numbers controlled by a different chip. These may not work until you load a driver for that set of ports, so try not to use them unless you plan to make sure they work for you.
Once you have the drive installed, boot into BIOS Setup and make sure the port you are using is Enabled, and that the BIOS "sees" the drive and identifies it correctly. IF this is an IDE device and port, skip down to the next paragraph. But IF this is a SATA drive, and IF you are using Win XP or some earlier OS, there is an additional setting you need to check. Look near the SATA Port Enabled setting for SATA Port Mode. The choices are usually IDE (or PATA) Emulation, Native SATA, AHCI, or RAID. For most systems running Win XP (or earlier), you should choose IDE (or PATA) Emulation. You see, Win XP and all previous Win's do not know how to use AHCI devices (that's the real nature of SATA drives) without having a driver installed. The easy way to avoid that is to have the BIOS fool Win XP into believing the drive is simply an IDE device it fully understands, and it will work. NOTE that this is NOT needed for Vista or Win 7. Both of those DO understand AHCI devices. So IF you are installing an old SATA drive and IF your OS is Vista or Win 7, set the SATA Port Mode to AHCI.
Lastly, check the Boot Priority Sequence to be sure it has not changed. I'm sure you do NOT intend to boot from this old drive, so it should not be anywhere in the Boot Sequence. Most often the sequence is optical drive first, then hard drive that contains the OS, and no other devices allowed. After making these changes, SAVE and Exit to boot, and your old HDD should show up in My Computer ready to use.