An easy way to remember this is to think of a machine on a server rack, we add more machines across the horizontal direction and add more resources to a machine in the vertical direction.
In the database world, horizontal-scaling is often based on the partitioning of the data i.e. each node contains only part of the data, in vertical-scaling the data resides on a single node and scaling is done through multi-core i.e. spreading the load between the CPU and RAM resources of that machine.
With horizontal-scaling it is often easier to scale dynamically by adding more machines into the existing pool - Vertical-scaling is often limited to the capacity of a single machine, scaling beyond that capacity often involves downtime and comes with an upper limit.
Good examples of horizontal scaling are Cassandra, MongoDB, Google Cloud Spanner .. and a good example of vertical scaling is MySQL - Amazon RDS (The cloud version of MySQL). It provides an easy way to scale vertically by switching from small to bigger machines. This process often involves downtime.
In-Memory Data Grids such as GigaSpaces XAP, Coherence etc.. are often optimized for both horizontal and vertical scaling simply because they're not bound to disk. Horizontal-scaling through partitioning and vertical-scaling through multi-core support.
You can read more on this subject in my earlier posts: Scale-out vs Scale-up and The Common Principles Behind the NOSQL Alternatives