What is the difference between a Relational and Non-Relational Database?
Most of what you "know" is wrong.
First of all, as a few of the relational gurus routinely (and sometimes stridently) point out, SQL doesn't really fit nearly as closely with relational theory as many people think. Second, most of the differences in "NoSQL" stuff has relatively little to do with whether it's relational or not. Finally, it's pretty difficult to say how "NoSQL" differs from SQL because both represent a pretty wide range of possibilities.
The one major difference that you can count on is that almost anything that supports SQL supports things like triggers in the database itself -- i.e. you can design rules into the database proper that are intended to ensure that the data is always internally consistent. For example, you can set things up so your database asserts that a person must have an address. If you do so, anytime you add a person, it will basically force you to associate that person with some address. You might add a new address or you might associate them with some existing address, but one way or another, the person must have an address. Likewise, if you delete an address, it'll force you to either remove all the people currently at that address, or associate each with some other address. You can do the same for other relationships, such as saying every person must have a mother, every office must have a phone number, etc.
Note that these sorts of things are also guaranteed to happen atomically, so if somebody else looks at the database as you're adding the person, they'll either not see the person at all, or else they'll see the person with the address (or the mother, etc.)
Most of the NoSQL databases do not attempt to provide this kind of enforcement in the database proper. It's up to you, in the code that uses the database, to enforce any relationships necessary for your data. In most cases, it's also possible to see data that's only partially correct, so even if you have a family tree where every person is supposed to be associated with parents, there can be times that whatever constraints you've imposed won't really be enforced. Some will let you do that at will. Others guarantee that it only happens temporarily, though exactly how long it can/will last can be open to question.
Hmm, not quite sure what your question is.
In the title you ask about Databases (DB), whereas in the body of your text you ask about Database Management Systems (DBMS). The two are completely different and require different answers.
A DBMS is a tool that allows you to access a DB.
Other than the data itself, a DB is the concept of how that data is structured.
So just like you can program with Oriented Object methodology with a non-OO powered compiler, or vice-versa, so can you set-up a relational database without an RDBMS or use an RDBMS to store non-relational data.
I'll focus on what Relational Database (RDB) means and leave the discussion about what systems do to others.
A relational database (the concept) is a data structure that allows you to link information from different 'tables', or different types of data buckets. A data bucket must contain what is called a key or index (that allows to uniquely identify any atomic chunk of data within the bucket). Other data buckets may refer to that key so as to create a link between their data atoms and the atom pointed to by the key.
A non-relational database just stores data without explicit and structured mechanisms to link data from different buckets to one another.
As to implementing such a scheme, if you have a paper file with an index and in a different paper file you refer to the index to get at the relevant information, then you have implemented a relational database, albeit quite a simple one. So you see that you do not even need a computer (of course it can become tedious very quickly without one to help), similarly you do not need an RDBMS, though arguably an RDBMS is the right tool for the job. That said there are variations as to what the different tools out there can do so choosing the right tool for the job may not be all that straightforward.
I hope this is layman terms enough and is helpful to your understanding.
Relational databases have a mathematical basis (set theory, relational theory), which are distilled into SQL == Structured Query Language.
NoSQL's many forms (e.g. document-based, graph-based, object-based, key-value store, etc.) may or may not be based on a single underpinning mathematical theory. As S. Lott has correctly pointed out, hierarchical data stores do indeed have a mathematical basis. The same might be said for graph databases.
I'm not aware of a universal query language for NoSQL databases.