[nl]

which problems a NoSQL database is useful for

It depends in which NoSQL database. Depending on your problem, you can probably find a NoSQL DB optimised for it. It will often be unclear if that NoSQL DB is actually better than a relational database until you try it.

Examples:

High write throughput: Cassandra

Simple key-value: Redis

Text search: Solr/Elastic Search

etc..

[twic]

Redis does more than simple key-value - rather than just reading and writing values, since values can be complex types like lists and dictionaries, it can insert into them, append to them, etc. It's still key-value, it's just not simple key-value!

That said, i would hesitate to describe Redis as a database at all. A key characteristic of databases is that they store every write in a durable way. Redis can checkpoint its state periodically, but as i understand it, it either can't or typically isn't used to safely keep every write. Redis is something in between a database and memcached. I doubt there's ever a situation where you have to choose between PostgreSQL/Cassandra/CouchDB and Redis; Redis is something you would use in addition to a database.

As for text search - RDBMSs have full text search, and at least in the case of PostregSQL, it seems pretty good - see slide 49 in http://es.slideshare.net/billkarwin/full-text-search-in-post... from 2009. You might not want to be leaning on your database for text search when you're at scale (for operational reasons more than performance ones), but it's a plausible way to start.