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: 
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).
[Netcraft](http://netcraft.com/) tracks market share of web servers & according to them, IIS stands second after Apache with a 26.03% share (as of July 2010). This is based on a survey of 205+ million sites

[This link](http://blogs.iis.net/bills/archive/2007/05/07/iis-vs-apache.aspx) has reasons why you should check out IIS7 if you use Apache today