Carbonic acid is a weak acid that is produced when carbon dioxide is dissolved in water. However, it's only slightly soluble so it can take some time to be produced (and this is why you can collect carbon dioxide over water as long as you're not working to exact quantities).
The formation works according to the following formula.
H2O + CO2 -> H2CO3
It is present in all fizzy drinks, as these are carbonated with carbon dioxide (!) and is somewhat responsible for the "sharp" flavour of these drinks. That said, many fizzy drinks have additional acid added as well (eg Coke has phosphoric acid added).
Carbonic acid formation can get in the way of certain quantitative analysis experiments in Chemistry. Phenolphthalein is a very sensitive acid/base indicator that can be used when performing a titration. If you have the known concentration acid in the beaker and dropping in the base from the burette, the solution will turn pink as soon as you cross the neutral point (interestingly, this indicator is pink with bases, not acids, as is more common). However, you have to be careful not to swirl the beaker around too much, as right on the transition, you could disolve enough carbon dioxide from the air to render the solution acidic again.
OldMiner asks "So, how does carbonic acid actually work as an acid? It must be a proton donor in some fashion to be an acid, yes? So what is the reaction that releases a free H+? The reaction you gives looks like it just combines happily with water, so there's no explanation for the pH change." And of course, he's right.
The chemical formula I have given above is the simple combination occuring. However, the acid partially disassociates into H+ and CO3- ions. The H+ ion then joins with another H2O molecule in the usual way to form a hydronium ion.