Big question.
Well, for starters, all the ones you listed there, generally speaking, are subspecies of one species,
Saccharomyces cerevisiae; not all yeasts are. Lager yeasts, for instance, are variously known as
Saccharomyces pastorianus or
Saccharomyces carsbergensis. When you get into more wild brewing, you get all sorts of exciting yeasts; one of my favourite wild ones is the
Brettanomyces genus, used in lambic beers and Orval; wild winemakers get fancy genuses like
Candide and
Zygosaccharomyces.
Enough Latin. The yeast is hell of important. A large flavour component in most beers and every wine comes from yeast flavours. Different strains of yeast are responsible for most (not quite all) fruity, spicy, bubblegummy, medicinal, yeasty, etc flavours you'll find in beer. A different yeast strain put into identical wort can make the difference between, say, a clove-and-banana-tasting Hefeweizen and a crisp clean American Wheat. You can get good descriptions of most every yeast strain available to homebrewers under the sticky "Ingredient Guides" in the ingredients forum.
As for the specific yeasts you mentioned. Ale yeasts are the standard beer yeasts (not lager) that ferment primarily at or around room temperature (60˚-70˚
. Coopers is one of these, marketed for beer, and you can get their style descriptions on one of the ingredient guides. The yeasts marketed for cider, while I've never used them, are I believe more closely related to wine yeasts. I don't know their flavour characteristics, but I bet you'll notice some sort of difference from Coopers. Wine yeasts come in a very wide variety, with a similar range of flavours to beer yeasts. I use a Montrachet yeast in my cider and Apfelwein, except when I'm feeling adventurous. Champagne yeasts are a type of wine yeast known to produce a very very dry finish, enough so that they've traditionally been pitched into just about anything that refuses to dry out.
So I guess you'll have to experiment, read up on things, and decide what you like. There's a lot out there.