Horizontal scaling refers to adding or removing databases in order to adjust capacity or overall performance. This is also called “scaling out”. Sharding, in which data is partitioned across a collection of identically structured databases, is a common way to implement horizontal scaling.
Vertical scaling refers to increasing or decreasing the performance level of an individual database—this is also known as “scaling up.”
It’s not free and application changes might be required (especially for horizontal scaling), but scaling with Azure SQL Database is pretty straightforward.