What are the advantages of using a schema-free database like MongoDB compared to a relational database?

Here are some of the advantages of MongoDB for building web applications:

1. A document-based data model. The basic unit of storage is analogous to JSON, Python dictionaries, Ruby hashes, etc. This is a rich data structure capable of holding arrays and other documents. This means you can often represent in a single entity a construct that would require several tables to properly represent in a relational db. This is especially useful if your data is immutable.
2. Deep query-ability. MongoDB supports dynamic queries on documents using a document-based query language that's nearly as powerful as SQL.
3. No schema migrations. Since MongoDB is schema-free, your code defines your schema.
4. A clear path to horizontal scalability.

You'll need to read more about it and play with it to get a better idea. Here's an online demo:

http://try.mongodb.org/

There are numerous advantages.

For instance your database schema will be more scalable, you won't have to worry about migrations, the code will be more pleasant to write... For instance here's one of my model's code :

class Setting
  include MongoMapper::Document

  key :news_search, String, :required => true
  key :is_availaible_for_iphone, :required => true, :default => false

  belongs_to :movie
end

Adding a key is just adding a line of code !

There are also other advantages that will appear in the long run, like a better scallability and speed.

... But keep in mind that a non-relational database is not better than a relational one. If your database has a lot of relations and normalization, it might make little sense to use something like MongoDB. It's all about finding the right tool for the job.

For more things to read I'd recommend taking a look at "Why I think Mongo is to Databases what Rails was to Frameworks" or this post on the mongodb website. To get excited and if you speak french, take a look at this article explaining how to set up MongoDB from scratch.

Edit: I almost forgot to tell you about this railscast by Ryan. It's very interesting and makes you want to start right away!

The advantage of schema-free is that you can dump whatever your load is in it, and no one will ever have any ground for complaining about it, or for saying that it was wrong.

It also means that whatever you dump in it, remains totally void of meaning after you have done so.

Some would label that a gross disadvantage, some others won't.

The fact that a relational database has a well-established schema, is a consequence of the fact that it has a well-established set of extensional predicates, which are what allows us to attach meaning to what is recorded in the database, and which are also a necessary prerequisite for us to do so.

Without a well-established schema, no extensional predicates, and without extensional precicates, no way for the user to make any meaning out of what was stuffed in it.

It's all about trade offs. MongoDB is fast but not ACID, it has no transactions. It is better than MySQL in some use cases and worse in others.

After a question of databases with textual storage), I glanced at MongoDB and similar systems.
If I understood correctly, they are supposed to be easier to use and setup, and much faster. Perhaps also more secure as the lack of SQL prevents SQL injection...
Apparently, MongoDB is used mostly for Web applications.
Basically, and they state that themselves, these databases aren't suited for complex queries, data-mining, etc. But they shine at retrieving quickly lot of flat data.