External Hard Drive size
Choose a reputable manufacturer: As data shows, HGST (formerly Hitachi, now owned by Western Digital) makes the most reliable hard-drives on the market today. Stay away from Seagate drives (see comments). 
Choose the right size drive: Regardless of the method, it's highly recommended to purchase an external drive that is atleast twice the capacity of the internal drive it’s backing up. 
Use Time Machine: There is no better local backup method for OS X. Alternatives exist in the form of archives (cloning your drive to another), but this is not a true, continuous backup. 


Sidenote: Networked backups using Apple’s Time Capsule is a reliable, wireless, way that is equally viable and definitely more convenient. The obvious con of this solution is the price-tag.  

Assuming your internal drive is no bigger than 500GB, here is the drive I would recommend: HGST Touro 1 TB 5400-RPM USB 3.0 External Hard Drive.



Updates: 


I own four HGST Touro drives. All have operated extremely well with no mechanical issues or problems to date. This is only my personal experience - which is all the more reason why it’s important to note that the Touro is the top-rated external hard drive on Amazon according to average customer reviews. 
@bmike brought up an important point. The Backblaze study on HDD failure rates used enterprise grade drives of 3.5’’ form factor. The consumer grade 2.5’’ external HDD’s, those relevant to the original question, were not the subject of these tests. See the comments below.