Vertical & horizontal wine tastings are designed to highlight differences between similar wines. In a vertical tasting, different vintages (e.g. 2007, 2006, and 2005 etc...) of the same wine varietal or style of wine from the same winery are tasted. This will emphasize the differences between various vintages. A vertical tasting is a fun way to learn about particular wines, as each vintage has distinct characteristics that set it apart from others.
While some wine critics will rave about one particular vintage; it doesn't mean that others do not have their merits or that a particular producer didn't make amazing wines in other years. So tasting the same wine over several vintages can be educational and enlightening in regards to the subtle, or not so subtle, difference from one vintage to the next.

 

It also gives you a chance to compare older vintages of a wine to younger ones, teaching you about how that wine ages and evolves with time.
Finally, because the wines are all made from the same vineyard and by the same winemaking choices and techniques (well within reason), you get to learn about that particular vineyard and the winemakers style.

In a horizontal tasting, the wines are all from the same vintage but are from different wineries. Keeping the wine variety or style and wine region the same, will help emphasize differences in winery styles, vineyard location, age of vines and the winemakers' style or personality.
There are many ways to design this theme but generally by sticking to wines with some similarity, generally wines from the same region and of course all from the same vintage. Whether you have 3 or 20 wines, try to create a range with some diversity within that region to compare different producers and vineyards.
A horizontal tasting can be easier to put together than a vertical wine tasting, particularly if you stick to a current, recent vintage which is still available in wine shops. While a vertical tasting of several vintages can be harder to find (because the older vintages are generally sold out or difficult to source).