Well, for a long time, "database" was synonymous with "SQL," with a few exceptions such as DBM-type systems, largely because there were a lot of them out there and they could do a lot of stuff. Two of the reasons the whole "NoSQL movement" got started are:

- RDBMSes were being used to do a lot of things that the relational algebra-based data model wasn't really designed for, like storing objects.

- A lot of the things that people were using RDBMSes for could be done more efficiently in a more specialized, somewhat lower-level product.

You can kind of compare "NoSQL" to using public and shared transportation instead of personal automobiles. Boats, buses, trains, and planes can't all do everything (at least, not efficiently), but the things they do well, they can do more efficiently than a car can.

So, basically NoSQL is about "relational databases aren't always the best choice, so use something that works well with your particular use-case." Though I usually prefer the term "non-relational" - the way "NoSQL" is used now suggests that they are some united front against relational DBs, but that's really not the case at all.