No, these terms don't refer to the respective positions you end in before and after wine tasting. Instead, horizontal and vertical are two different types of tastings, or comparing wines by sampling them. The best way to judge wines is to compare them side-by-side, and so, a couple of different methods have developed to facilitate the best way to do that.
There are many different ways to organize wines for a tasting. You can, for example, choose wines in a certain price point or from a certain part of the world. But if you want to taste like the pros, you'll decide on either a vertical tasting or a horizontal tasting.
Vertical wine tastings are best if you want to learn about a particular wine producer and the style of their wines. It involves tasting several wines from the same winemaker. You can take a variety of wines, but most vertical wine tastings limit themselves to one wine from several different vintages. This is a great way to learn a lot about wine quickly, because it allows you to compare wines that are identical except for the vintage. Since the only differences between the wines will be their year, what you'll learn in a vertical wine tasting is how greatly the growing conditions affect the final product. The subtle differences between vintages really become apparent in a vertical wine tasting, and you'll probably learn that the variation from vintage to vintage is actually much larger than you thought. Vertical wine tastings help develop your palate by teaching you to pick out subtle differences between wines.
To host your own vertical wine tasting, all you need is a few different vintages from the winery of your choice. An ideal vertical tasting would have one bottle of every vintage every made by a certain producer, but this isn't usually practical or even possible. Realistically, there are no limitations on the number of bottles you should taste, but since the more you sample, the more you learn, you might want to call over some friends to help you. Serve the wines in chronological order from youngest to oldest, so that your palate moves from the simpler young wines to the more complex and subtle flavors of the old ones.
Horizontal wine tastings are a great way to learn about the differences between producers. In a horizontal wine tasting, you taste several different wines from the same vintage. The wines usually have something more in common as well. Usually, the wines in a horizontal wine tasting come from the same region. If they don't, the characteristics can be so different that you really won't get much out of tasting them. But you don't have to be limited by region. A more casual horizontal tasting could include just white wines, wines from a single variety of grape, or whatever parameters you choose, as long as each bottle is the same vintage.
Wine tastings are a great event to do with a group, whether its a small and intimate one or a loud, large party. They make the perfect ice breaker, giving your guests an instant topic of conversation. And wine tastings always tend to be full of fun and laughter. By making your basic tasting into a vertical or horizontal tasting, you give yourself and your guests the opportunity to really get something out of the tasting by learning more about the wine and educating your palate.