Hard disk drives (HDD) are spinning mechanical platters that read/write info via magnetic patterns on the platters. HDDs vary by storage capacity ( ie 500MB, 1TB, 2TB, 3TB and now 4TB). and by performance. The main things that contribute to performance are: rotational speed (5400, 7200, 10000) where faster is usually better, cache size (small amount of memory on the hard disk's controller) where larger is usually better, and external interface (SATA 3GB, SATA 6GB) where the newer SATA 6GB is a larger/faster pipe.
A solid state drive (SSD) looks to the rest of the computer like a hard disk. But instead of spinning magnetic platters, it uses memory chips. This gives an SSD much faster read/write times compared to traditional spinning platters. The down side of SSD......cost.
Time Machine is a software program that comes with Macs. Its job is to backup your Mac to the hard disks you specify. Typically the backup drive(s) is external to the Mac. It can be connected to the Mac directly via USB 2, FW, USB 3, thunderbolt. The back drive can also be connected via ethernet or wifi when the drives is connected to a remote device. You do want to set up a backup drive and have Time Machine do routine backups of your Mac. Personally I use Time Capsule for my main backup. Time Capsule is a white box that does several things: contains a hard disk drive (2TB or 3TB) for backing up your Mac, an IP switch for connecting your Mac(s) and printers to a broadband (cable or DSL) modem, and a USB 2 port for connecting a printer or external HDD. And last, but not least, the Time Capsule is a wifi receiver that lets you set up wifi in your home.
