Q:

Am I missing something about Document Databases?

I've been looking at the rise of the NoSql movement and the accompanying rise in popularity of document databases like mongodb, ravendb, and others.  While there are quite a few things about these that I like, I feel like I'm not understanding something important.
Let's say that you are implementing a store application, and you want to store in the database products, all of which have a single, unique category.  In Relational Databases, this would be accomplished by having two tables, a product and a category table, and the product table would have a field (called perhaps "category_id") which would reference the row in the category table holding the correct category entry.  This has several benefits, including non-repetition of data.  
It also means that if you misspelled the category name, for example, you could update the category table and then it's fixed, since that's the only place that value exists.
In document databases, though, this is not how it works.  You completely denormalize, meaning in the "products" document, you would actually have a value holding the actual category string, leading to lots of repetition of data, and errors are much more difficult to correct.  Thinking about this more, doesn't it also mean that running queries like "give me all products with this category" can lead to result that do not have integrity.
Of course the way around this is to re-implement the whole "category_id" thing in the document database, but when I get to that point in my thinking, I realize I should just stay with relational databases instead of re-implementing them.  
This leads me to believe I'm missing some key point about document databases that leads me down this incorrect path.  So I wanted to put it to stack-overflow, what am I missing?

A:

You completely denormalize, meaning in the "products" document, you would actually have a value holding the actual category string, leading to lots of repetition of data [...]

True, denormalizing means storing additional data. It also means less collections (tables in SQL), thus resulting in less relations between pieces of data. Each single document can contain the information that would otherwise come from multiple SQL tables.
Now, if your database is distributed across multiple servers, it's more efficient to query a single server instead of multiple servers. With the denormalized structure of document databases, it's much more likely that you only need to query a single server to get all the data you need. With a SQL database, chances are that your related data is spread across multiple servers, making queries very inefficient.

[...] and errors are much more difficult to correct.

Also true. Most NoSQL solutions don't guarantee things such as referential integrity, which are common to SQL databases. As a result, your application is responsible for maintaining relations between data. However, as the amount of relations in a document database is very small, it's not as hard as it may sound.
One of the advantages of a document database is that it is schema-less. You're completely free to define the contents of a document at all times; you're not tied to a predefined set of tables and columns as you are with a SQL database.
Real-world example
If you're building a CMS on top of a SQL database, you'll either have a separate table for each CMS content type, or a single table with generic columns in which you store all types of content. With separate tables, you'll have a lot of tables. Just think of all the join tables you'll need for things like tags and comments for each content type. With a single generic table, your application is responsible for correctly managing all of the data. Also, the raw data in your database is hard to update and quite meaningless outside of your CMS application.
With a document database, you can store each type of CMS content in a single collection, while maintaining a strongly defined structure within each document. You could also store all tags and comments within the document, making data retrieval very efficient. This efficiency and flexibility comes at a price: your application is more responsible for managing the integrity of the data. On the other hand, the price of scaling out with a document database is much less, compared to a SQL database.
Advice
As you can see, both SQL and NoSQL solutions have advantages and disadvantages. As David already pointed out, each type has its uses. I recommend you to analyze your requirements and create two data models, one for a SQL solution and one for a document database. Then choose the solution that fits best, keeping scalability in mind.

A:

I'd say that the number one thing you're overlooking (at least based on the content of the post) is that document databases are not meant to replace relational databases.  The example you give does, in fact, work really well in a relational database.  It should probably stay there.  Document databases are just another tool to accomplish tasks in another way, they're not suited for every task.
Document databases were made to address the problem that (looking at it the other way around), relational databases aren't the best way to solve every problem.  Both designs have their use, neither is inherently better than the other.
Take a look at the Use Cases on the MongoDB website: http://www.mongodb.org/display/DOCS/Use+Cases

A:

A document db gives a feeling of freedom when you start. You no longer have to write create table and alter table scripts. You simply embed details in the master 'records'.  
But after a while you realize that you are locked in a different way. It becomes less easy to combine or aggregate the data in a way that you didn't think was needed when you stored the data. Data mining/business intelligence (searching for the unknown) becomes harder.
That means that it is also harder to check if your app has stored the data in the db in a correct way. 
For instance you have two collection with each approximately 10000 'records'. Now you want to know which ids are present in 'table' A that are not present in 'table' B. 
Trivial with SQL, a lot harder with MongoDB.
But I like MongoDB !!  

