The first thing confusing me is alkalinity.
It's supposed to. Otherwise they wouldn't use the foolish units they do. Alkalinity is actually a very simple thing. It is the number of mL of 0.1 N acid that must be added to 0.1 L of the water sample to lower its pH to 5.5 at which pH almost all the carbonate and bicarbonate are converted to carbon dioxide. Since 0.1 N acid contains 0.1 mEq/mL the number of mL used to acidify 0.1 L us the the number of mEq/L. If we stopped there everything would be easy but in the US we multiply the mEq/L by 50 because if you take 100 mg of calcium carbonate (1 mmol) and dissolve it in 1 L of water
using carbonic acid as the required acid and bring the pH to 8.3 you will have, approximately 2 mmol/L bicarbonate ion each of which requires a mEq of acid to convert to CO2. Thus water in which 100 mg of limestone was dissolved per liter by carbon dioxide has alkalinity of about 100 ppm 'as CaCO3' and the same calcium hardness.
This just lists "Alkalinity ppm". What unit is this?
In nearly all cases it is ppm as CaCO3. Just divide by 50 to get mEq/L.
All I know about water is the bit I've read in "How to brew", and if I understand it correctly, it should be either as CaCO3 or as HCO3.
It should be mEq/L but in the us it will be as CaCO3 in nearly all cases. It can be expressed as ° dH (based on calcium oxide) or bicarbonate or really anything you want. The only advocate of 'as bicarbonate' is the author of Bru'n water which is a pity as it mars a product which otherwise has lots to offer.
Is there any way to tell which?
In a good report it will be clearly labeled. Sometimes it is labeled as just carbonate and you have to be particularly careful with, for example, posts here where people misunderstanding what is really involved will often say 'my carbonates are 120' when the fact is the alkalinity is 120 ppm as CaCO3. If there is a question about the labeling usually the numbers give a clue. Numbers of less than 10 would usually indicate mEq/L (unless the hardness is a commensurately small number). The only place you are likely to see alkalinity represented as bicarbonate is in Bru'n water. The magnitude of alkalinity values as ppm bicarbonate is close to the representation as calcium carbonate (61 vs 50 for 1 mEq/L). Ultimately, you can take the numbers and put them into a spreadsheet which calculates all the anions and cations and sums them. In the real physical world that sum will be 0. The unit choice for alkalinity which gives the sum closest to 0 is the right one.
Yes, it matters a great deal as alkalinity is the most important water parameter because of its effect on mash pH.
There's also the ranges. What do I use in a calculator? The max? The middle of the range?
The value you want is the one for the water you are brewing with on a particular day. If you have a wide variability over the short term you should either measure the alkalinity (easily done with simple kits) or run the water through an RO filter thus removing the alkalinity and the variability.
Or should I just send a sample out to be tested? If so, what's a good place to use for brewing purposes?
Everyone here uses Ward Labs. They offer a very good test for the price.