Do not confuse a NoSQL type database with something like memcached (which is not intended to store data permanently).
Typical use for memcached is to store some query results that can be accessed by a cluster of web servers - ie. a shared cache. Eg. On this page is a list of related posts and there is likely a bit of work for the database to do to produce that list. If you do that every time someone loads the page then you will create a lot of work for the database. Instead, the results once retrieved for the first time could be stored on a memcached server with the key being the page ID. Any of the web servers in the cluster can then fetch that information very quickly without having to constantly hit the database. After a while, the cache entry would be purged by memcached so that the results for old articles don't use up space. [Disclaimer: I've no idea if StackOverflow does this in reality].
A "NoSQL" database on the other hand is for storing information permanently. If your data schema is quite simple and so are your queries, then it may be faster than a standard SQL database. A lot of web applications don't need hugely complex databases, and so NoSQL databases can be a good fit.