Apache is still the most popular web server on the Internet, estimated to serve about 55% of all websites in existence, compared to nginx's 12% (in Jan 2013).
UPDATE (Apr 2015): Apache still has the largest share at about 40%, and nginx has grown to 15%. Check the above mentioned link for the latest figures.
UPDATE (Jul 2020): nginx has comfortably surpassed Apache in popularity at 37% vs 25%. Apache still has more hits on searches.
It is the oldest web server around, which means that you won't have any trouble finding people skilled in configuring it.
You're also slightly more likely to find help when searching the web. SO has ~80,000 tags for Apache, and ~40,000 for nginx.
Installation documents sometimes assume you're using Apache, often including .htaccess documentation, but not the equivalent nginx configuration:
My recommendation? Either are perfectly good choices, flip a coin if you can't find any compelling reasons either way.
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