What is the difference between a Relational and Non-Relational Database?

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.

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 hav