The term splog comes from combining the words Spam and Blog.
Splogs are fake blogs that spammers setup to link to their other sites and/or get ad impressions (often AdSense) from visitors. The term was popularized around mid August 2005 when it was first used by some high profile bloggers but appears to have been used a few times before for this purpose.
Splogs are a real problem because they load up search results with fake and useless sites, just like most other forms of spamdexing. The difference here is the spammers are usually using free blog services to setup their blogs.
As such, many blog related services are working on ways to indentify splogs and remove / ignore them. IceRocket, a blog search engine, claims to be a big backer of this approach, and work hard to weed out splogs from their index.
Some large blog hosts have been reluctant to remove blogs because they wanted to protect freedom of speech or something like that.
Until recently 'blogger' (a.k.a. blogspot) was failing to take any action against splogs, causing us to list them as LoserWebmasters. This is ironic since they are owned by Google, who's search engine service is hurt by this spamdexing.
Joe indentified a few examples of spam blogs a long time ago, but it seems the situation got a lot worse, and blogger became more splog than blog. See this comic strip
The publicity of the problem finally meant they acknowledged it and implemented a flagging system built into the Blogger navigation bar that allows visitors to mark objectionable blogs for review.
Blogger is decreasing the visibility of the splogs quickly by removing them from the Next Blog ring, but they are much slower in actually removing them from their service. Shortly after implementing the flag, the Next Blog ring is nearly clear of spam, but most splogs flagged by spam fighters are not being removed.
There is actually a new wiki for collaborating on splog flagging efforts.
The primary purpose of splogs is links for increasing PageRank of thier real sites. Based on the number of AdSense ads on most splogs and similar spamdexing sites, getting actual human hits is certainly high on their list.
It didn't used to be this way though. Outside of splogs it is still common for doorway spam pages not to have ads. This means getting splogs removed from Blogger's Next Blog ring doesn't really affect the first purpose much.
Another likely reason for the popularity of splogs is spammers believe that blogs get better search ranking just based on their format. In reality that isn't true. The reason blogs are often ranked high for search terms is because of the way bloggers interlink with other blogs and post frequently on related topics. It has nothing to do with the blog format.
Some people worry their blogs are going to be targeted in the attack on splogs. Don't worry. There are obvious differences between splogs and legitimate blogs. Splogs are only there for the purpose of driving visitors to other sites, getting ad (especially AdSense) impressions, and/or using the links for increasing PageRank of the other sites. Most have little or no content and if they do it is usually nonsense. So far most usually are using default themes and no graphics. And posts on them are often made by an automated program.