What does horizontal scaling mean?

Horizontal Scaling
Horizontal Scaling is essentially building out instead of up. You don't go and buy a bigger beefier server and move all of your load onto it, instead you buy 1+ additional servers and distribute your load across them.

Horizontal scaling is used when you have the ability to run multiple instances on servers simultaneously. Typically it is much harder to go from 1 server to 2 servers then it is to go from 2 to 5, 10, 50, etc.

Once you've addressed the issues of running parallel instances, you can take great advantage of environments like Amazon EC2, Rackspace's Cloud Service, GoGrid, etc as you can bring instances up and down based on demand, reducing the need to pay for server power you aren't using just to cover those peak loads.

Relational Databases are one of the more difficult items to run full read/write in parallel.

I saw Damien Katz speaking about CouchDB at StackOverflow DevDays in Austin and one of his primary focuses for its creation was these parallel instances. As this has been a focus of it since day one, it would be much more capable of taking advantage of horizontal scaling.

Vertical Scaling
Vertical Scaling is the opposite, build up instead of out. Go and buy the strongest piece of hardware you can afford and put your application, database, etc on it.

Real World
Of course, both of these have their advantages and disadvantages. Often times a combination of these two are used for a final solution.

You may have your primary database where everyone writes to and reads real time data on a large piece of hardware. Then have distributed read only copies of the database for heavier data analysis and reporting where being up to the minute doesn't matter as much. Then the front end web application may be running on multiple web servers behind a load balancer.