Relational databases are based on relational algebra, which is more or less an outgrowth of set theory. Relationships based on set theory are effective for many datasets, but where parent-child or distance of relationships are required, set theory isn't very effective. You may need graph theory to efficiently design a data solution. In other words, relational databases are overkill for data that can be effectively used as key-value pairs and underkill for data that needs more context. Overkill costs you scalability; underkill costs you performance.
Key-value pair databasesKey-value pair databases include the current 1.8 edition of Couchbase and Apache Cassandra. These are highly scalable, but offer no assistance to developers with complex datasets. If you essentially need a disk-backed, distributed hash table and can look everything up by identity, these will scale well and be lightning fast. However, if you find that you're looking up a key to get to another key to get to another key to get to your value, you probably have a more complicated case.
There are a number of different permutations of key-value pair databases. These are basically different trade-offs on the CAP theorem and different configurations of storage and memory use. Ultimately, you have some form of what is basically a hash table.
This is fine for flat parts lists so long as they don't composite. This is also fine for stock quotes, "right now," or other types of lists where that key has meaning and is the primary way you're going to look up the value. Usually these are combined with an index, and there is a way to query against the values or generate a list of keys, but if you need a lot of that, you probably should look elsewhere.
Column family/big table databasesMost key-value stores (including Cassandra) offer some form of grouping for columns and can be considered "column family" or "big table" as well. Some databases such as HBase were designed as column family stores from the beginning. This is a more advanced form of a key-value pair database. Essentially, the keys and values become composite. Think of this as a hash map crossed with a multidimensional array. Essentially each column contains a row of data.