Not too sure how to answer this directly, so I'll give some sort of database history explanation :P
Historically seen, there have been a few types of databases. The most important ones being hierarchical, a network database or a relational database. This is also more or less the following order in which they appeared, but the first two ones are considered to be "legacy software".
Now every database needs a "language" to work with the data. This is called a "Data Manipulation Language", and SQL is an implementation of such a language. More specific, SQL is the DML for most relational databases. A lot of vendors of relational databases choose SQL as their DML because it's easy to work with and because people already know it. But it's also pretty specific to relational databases. So you won't find it easily for another type.
And of course, relational databases have some characteristics. For example: before you can store any data, you have to define what data you want to store, in what format it will be and so on. In other terms, you have to create tables and columns first, and define the links between these tables.
NoSQL is just a collection of other types of databases, so not relational databases. It stands for "not only SQL". It often has the advantage that you can set up a DB and you can use it straight away. Take mongodb as an example. You could just make an object and then tell mongoDB: "Yo, store this thing for me please". There's no defining the data beforehand involved, so you could pretty much store whatever you want.
So it's all about the use case. Relational databases are really good for storing data that's very structured and where you know beforehand how it will look like. Sometimes you need something else though, and these types of databases will often not use SQL. I gave the example of mongoDB already. Another example would be redis, which is an in-memory DB. This means it's really fast, but can't store as much as a traditional, relational DB.
Yet another example, the other extreme if you will, is hadoop, which allows to create giant clusters to form one big database.
So no, not all databases are tables, rows and columns. Some databases store their data internally in a different way so that it's easier to work with in certain use cases.
I hope this cleared things up a bit.