It's easier than you think it is. First, to get a general explaination of how Time Machine works and how you can use it, see > http://support.apple.com/kb/HT1427?viewlocale=en_US&locale=en_US
Time Machine needs that your external drive meets some requirements. You will see them when you select the Time Machine drive, if your external drive doesn't meet requirements.
Apart, if you want more information, see > http://pondini.org/TM/Works.html I recommend you to store that website with a bookmark: that site will solve most of your problems and questions
You need both because Time Machine is the software that backs up your internal hard drive to an external hard drive. Backing up to the same hard drive would be superfluous since you couldn't restore the backups from the dead hard drive.
When you plug in the hard drive, it should ask you if you want to use that disk for Time Machine; click yes and it will format the drive for Time Machine and turn on Time Machine. The first backup will take a while as it is copying everything. Then, every hour, Time Machine will back up to that disk if it is connected. You don't have to have it plugged in all day. Just connect it at the end of the day and let it do it's backups. After a few days, Select Enter Time Machine from the Time Machine icon in the Menu bar. It will go into it's starfield display and make sure you can scroll through the days with th slider on the right side. If you can, it is making backups and you can let it do its thing.