Databases that do not fall into a neat grid. You don't have to specify which things are integers and strings and booleans, etc. These types of databases are more flexible, but they don't use SQL, because they are not structured that way.
The advantage of using a NoSQL database is that you don't have to know exactly what your data will look like ahead of time. Perhaps you have a Contacts table, but you don't know what kind of information you'll want to store about each contact. In a relational database, you need to make columns like "Name" and "Address". If you find out later on that you need a phone number, you have to add a column for that. There's no need for this kind of planning/structuring in a NoSQL database. There are also potential scaling advantages, but that is a bit controversial, so I won't make any claims there.
Disadvantages of NoSQL databases is really the lack of SQL. SQL is simple and ubiquitous. SQL allows you to slice and dice your data easier to get aggregate results, whereas it's a bit more complicated in NoSQL databases (you'll probably use things like MapReduce, for which there is a bit of a learning curve).