To scale vertically means that you replace your garage machine, that your cat is using to play Sudoku, by a mid-range server, that may soon became too underpowered, until you have to replace him by an ultra-fast and hyper-expensive high-end server. The point is that you can't extend your server collection but have to replace slow calculators by faster ones.
To scale horizontally means to assign more servers to the same task. It's basically the idiomatic cloud computing concept.
Striving for the later is better in most cases. An economical advantage of horizontal scalability is that it might be cheaper to buy many mid-range machines than a single high-end monster.