I was asked by a gentleman called Fred whether a laptop hard drive can be plugged into a desktop PC? "The pin out appears to be the same," Fred said, "but there is no separate power-supply connector."
Short answer: yes. Long answer: it depends on a few things. For starters, Fred, you neglected to tell me what kind of drive it is - IDE or SATA - or what kinds of drive(s) you have in your desktop.
However you did give me a clue. Because there's "no separate power-supply connector," that means it's most likely a 2.5-inch IDE drive. (Newer SATA drives for laptops have the same interface and power connectors as their larger desktop counterparts.)
If I'm right, you can use your laptop drive inside your desktop - but you'll need an adapter. Fortunately, they're cheap; the Rosewill RCW-616 Laptop 2.5" to Desktop 3.5" IDE Hard Drive Adapter Converter is on Amazon for around £5. It includes the necessary Molex adapter for connecting to your desktop power supply.
However, your desktop might not have a drive bay that can accommodate a 2.5-inch drive. If that's the case, there are adapters available.
On the other hand, consider an easier and more versatile solution: an enclosure that turns your old internal laptop drive into an external USB drive. That not only saves you from having to tinker around inside your desktop, but also gives you a drive you can easily use with multiple PCs.