There are several options for adding extra storage to a computer or network of computers. The simplest of these is an external hard drive. There aren't too many considerations to take into account for a single drive solution. The real choice is in what interface you use,either USB or eSATA. USB has more support but you will generally get better speeds from eSATA, at a slight price premium. USB 3.0 theoretically should level the playing field but for that you'd need a compatible motherboard and enclosure. Beyond the choice of interface, the options are to buy an external hard drive from a name manufacturer or buy an internal hard drive and enclosure to make your own. I've never really been in the market for single external hard drives, when I've used external drives they've been old drives that I upgraded from. I believe that enclosure+internal drive gets you more storage per dollar over external.
When buying hard drives, www.newegg.com is always my first choice because the reviews of available drives give a good picture of which drives could be problematic. There is a sweet spot for storage that's usually one or two notches down from the largest drives on the market, right now its 2 TB. Performancewise, big drives are generally better, featuring more modern tech than smaller drives although starting around the 2TB mark some drives have a new organizational system that can cause compatibility problems with older hardware.
For enclosures, I've had good results with Vantec, also bought from newegg. If portability is more of an issue you can also go with 2.5" laptop drives and smaller enclosures.
The next step up from the single drive system is to get a dedicated RAID enclosure. You can get 2 drive solutions but in order to get RAID 5 you need at least a four drive enclosure. Moving from 4 drive bays to 5 comes at a premium so unless there's a good deal or you really need the extra space 4 will work fine. With a multibay enclosure the options are to get a NAS, which operates by itself on a network, or a direct attached storage box (typically via eSATA). One thing to bear in mind is that with a RAID system, the Green drives that are usually pretty cheap can be problematic. Going with enterprise drives isn't necessary but pay close attention to reviewer comments and message boards in choosing drives for a NAS system.
Several years ago there wasn't much competition for consumer grade NAS boxes and Infrant's READYNas were the best option. Since then, Infrant got bought out by Netgear, resulting in an inferior lower tier line and a significantly more expensive high performance line. Other competitors have stepped up though and affordable, high performance NAS boxes are available. Smallnetbuilder.com is an excellent resource for keeping up to speed on which NAS boxes are the best deal.
With direct attached storage the options tend to be simpler as the extra features such as built in servers and torrent clients aren't there. Newegg provides a decent product selection.
The last option is cloud storage. This option is really meant for lesser amounts of storage but does benefit from much more mobility. Dropbox is a good choice, with 2GB free but the possibility to expand that with a little effort. I only started using Dropbox a little while ago and its a very useful tool.