Question: I've traditionally deployed my sites under apache, simply by default. I've heard things about NGinX though and I'm wondering under which conditions it would be superior. In addition to apache and Nginx, what the other options are for webservers and what are their benefits?

Answer: 
Apache can be a bit heavy, and can fall over when under heavy load. There are lighter weight servers available, including nginx, lighttpd, and cherokee.

These lightweight options can be set up to serve static files quickly, and delegate dynamic server-side work to Apache.
There are a PILE of webservers out there. I expect depends on the usage you want. Apache will do most of what people want, it's known good, well updated, and will run on nearly any hardware and operating system combination.

For a comparison of web servers, check [Comparison of web server software](http://en.wikipedia.org/wiki/Comparison_of_web_server_software).