You have to consider three main topics when choosing an external hard drive: the size, capacity and connection to the computer.
External hard-drives come mainly in two different sizes: 2.5 and 3.5 inches (being that dimension the width of the internal drive). Size means weight, of course.
So, if you intend to carry the drive all around, you should look for smaller ones, so that weight does not become a drawback.
In the market, you will also find some "exotic" smaller drives, in 1.8 in size, for example.
However, those are usually more expensive. If you are not looking for "extreme" portability, you should stick to 2.5in. drives and the price will also be much fairer.
The types of drives and their capacity
Logic data (bits and bytes) are stored in the internal plates' surface. A minimum surface is needed per bit, so it is logical to observe that bigger drives may offer bigger capacity.
With latest state-of-the-art drives, this is usually true. Current 2.5in. drives reach up to 1Tb (1,000 Gb), whereas 3.5in ones can go past 2Tb.
If you will work with many files or big ones, you may need a 3.5in external storage. It may lose its portability, but it will make a perfect companion for your desktop computer.
Everyday data and the usage of the drive
If you just need to carry your everyday data (accounting, text files, latest pictures to show, a couple of movies.) 2.5in. drives will already do good.
Again, latest launches (both in 2.5 and 3.5in formats) will have a penalty cost. Hence, consider if you need the biggest drive, or if you can survive with a (cheaper) 500Gb drive.
Connection to the computer
External hard disks
Finally, you have to select an external hard drive that will be recognised by your computer.
Currently, the de-facto standard is USB connection, in its 2.0 revision. Now, you can already buy some USB3.0 drives (at a cost) that offer higher transmission rates. They are good for big files or may give plenty of space for-smaller ones.
Alternatively, you may find some other interfaces in the market, like Firewire (now quite limited to Mac computers) and eSATA, an extension of the SATA internal interface for external devices. In this case, ESD protection is added to protect both computer and drive.
One last remark: modern 2.5in. drives do not require an external power adapter . They will use power from the USB port. This is also useful if you need to take the portable storage with you...