Are there any breweries that have a distinct "house flavor" that is present in most of their beers to you? For me, Unibroue is a prime example. I can pick out their house character in pretty much all their beers. Sam Smith is another, and Harpoon as well (for at least a few of their beers) for me. That "house character" is almost always from their yeast strain. On the highest level, that is how I put yeast into context from a flavor standpoint.
Now, you can affect the amount and/or balance of certain flavors produced by the yeast by adjusting various factors. Fermentation temperature is the biggest thing, followed closely by pitching rate, oxygenation and wort gravity. Each yeast strain responds differently or in different degrees to these factors, but the underlying signature is almost always there. For example, when I brew with Wyeast 3864 (Canadian/Belgian - the Unibroue strain), my beers have that unmistakeable "Unibroue" character.
As far as which strain does what, I'd read some of the descriptions on the yeast labs' websites. In general, you want to select a strain similar to the beer you want to brew (English Ale strain for an English Ale, German Lager strain for a German lager, etc). I encourage you to experiment. Try splitting a batch into a few smaller fermenters and pitch different yeast strains in each to compare and contrast. Another good idea is to take one strain and brew several different beers with it so you can learn how it responds to different variables.