SSD is "Solid State Drive". It is a special form of RAM together with its own controller in a package the same size as a conventional hard drive, with the same connectors. It is made to perform in many ways exactly like a hard drive. BUT since it has NO moving parts, it is VERY much faster than "normal" mechanical HDD units. AND very much more expensive per GB capacity. Generally you cannot find (or afford) SSD's as large as the largest HDD units. The optimum use for SSD's these days is as an internal drive that holds the OS and key applications software that MUST have fast performance. DATA and lesser app software is placed on a second internal HDD of the conventional mechanical type.
Hard drives come in two main physical sizes (plus a few others). The most common are those in desktops machines, called 3½" drives (their width). The others are smaller 2½" drives, commonly used in laptops. Because the latter are smaller and sell fewer, they tend to be more expensive per GB, and have smaller maximum capacities. For your use I suspect the 3½" size is what you want. Either can be inside an external drive unit.
External drives that do NOT need their own power supply module draw all their power from one or two USB ports. However, USB2 ports have a limit on how much power they can provide, so almost all of these units are based on smaller 2½" HDD's inside. Most units with 3½" HDD's have their own power supply, either as a "wall wart", as a box in the middle of a cable, or built into the case with only a plain power cord to the wall. I prefer not to have the power supply inside the case - it eliminates one internal source of heat.
You can buy complete external drive units in various configurations. The straightforward ones are just one HDD unit in an enclosure with a power supply system and one or more ways to connect to the computer. More complex ones may include (or have an expansion slot for) a second HDD unit inside, and some way to use two HDD's as a RAID array or a JBOD array.
The external interface - how your external drive connects to the computer - may be USB2, eSATA, or Firewire 400 (aka IEEE 1394a). USB3 is just coming on the scene. Firewire 800 (aka IEEE 1394b) is not common in PC's, but seen on many Macs. Of these, USB2 is the slowest, eSATA is often twice as fast (may be just as fast as an internal SATA drive), Firewire 400 is in between those. USB3 and Firewire 800 both MAY be faster than eSATA, but maybe not - depends on limits imposed by the HDD itself, not by the interface. It is VERY common these days to find externals with two interfaces (e.g., USB2 plus eSATA), or even three, but you only get to use ONE of these at a time. Choose your unit by the ports you have on your computer. I have an eSATA port on the mobo, so I bought a unit with both USB2 (almost universally available on other people's computers) and eSATA, and use the latter.
USB2 and USB3 both have power available to the attached device in the computer port, but limited - USB3 has a higher power limit. eSATA generally does NOT have power in the port, but some new extensions of that system do add power on two pins. Personally, I prefer to have an external drive with its own power supply anyway to avoid loading the main computer.
Many ready-to-use external drives come with free software packages for things like backups, etc. Depending on what you plan to do, and on reviews of that free software, these may be a good feature or not.
A good alternative for some people is to assemble your own external drive by buying separately an enclosure and a HDD. When you do that, make sure you buy one that matches the HDD in two ways:
(a) the drive size - 3½", 2½", or something else; and,
(b) the HDD's own interface - IDE or SATA. (The latter is likely to be SATA II (more properly, SATA 3.0 Gb/s), and there is no speed advantage in SATA 6.0 Gb/s (unless you're buying a SSD) so I would not pay extra for that. The internal connection for the HDD unit itself is completely separate from the external interface to the computer. However, it happens that one of the fastest combinations is an eSATA connection to the enclosure, and a SATA 3.0 Gb/s HDD inside. This system basically requires no "translation" of data inside the enclusure, so it can be fast and efficient.
Extra features to consider:
(a) Power supply system - outlined above;
(b) Fan or not in the case? A fan will cost more but keep your HDD inside cooler. I worry about when it will wear out. I prefer a fanless case of metal with good cooling slots for airflow, and a low-power HDD unit inside.
(c) On / Off switch on the case?
(d) Indicator light on the case?
When you buy your own enclosure and drive, you don't usually get free software, but you can dig that up yourself if you want.
For the HDD, you get to choose exactly which unit you put inside - your preferred manufacturer and model / size. Some people, for example, would choose a WD Green line drive in the 1 to 2 TB size. They are reliable, low power consumers (low heat generation), and slower than the WD Black line. But for backups and media storage / retrieval, the slightly slower average data transfer speed does not matter. It is reported that Green drives on an eSATA interface are fully capable of supplying data for 3D HD movie playback. This may NOT be ideal for gamers who demand the ultimate performance. Often the price per GB is best around 1.5 TB, maybe 2.0.