laptop hard drives fit in a caddy. You will need to power down your laptop, and unscrew the the caddy from the case. gently remove the caddy with the HD on it. Some makers like Toshiba also have an adapter that plugs into the pins on a hd. IF so, gently remove it. You will need it later.
When shopping for HDs for a laptop, speed is important. Most HDs on laptops are unbearably slow. Get a 7200 speed or faster. you will need to specifically shop for a LAPTOP HDs. Note the Size of your HD. Generally they're 2.5 in physical size.
the next important consideration is capacity. Since few laptops allow more than 1 internal HD, get a big HD. You can later partition this drive into small partitions if need be.
Faster hard drives also improve battery life when using batteries. Since they read faster, they access less thereby using less juice. You won't notice much of an improvement, but there is a tad improvement. To really boost battery life and performance overall, add RAM instead.
Generally you shouldn't have any issues with 3rd party brands, just remember, you will need to connect the adapter (if any) and the tray onto your new hard drive before sliding the tray and re-screwing it back in.
Once you have completed this operation, format and install your OS and files as normal.
There are some laptop hard drive makers that have converter kits. You basically clone your OS and files to your new HD before installing the HD. You then can use your old HD as an external, spare drive. These are nifty and I have one.
Ensure when you slide the caddy with your new HD into the computer to do it gently and not to bend the pins. USE the adapter if it has one. If you don't it won't work and in the worse case sceario, you'll bend or break your pins on your new HD. very bad.
good luck!