Most municipal water is treated with chlorine or chloramine to kill bugs. This helps to guarantee that the stuff that comes out of your tap won't kill you. The problem for brewers is twofold: deadly organisms can't survive in beer anyway, and chlorines can do bad things to the flavor of your beer.
Chlorine isn't so bad. Chlorine is volatile, and will leave your water pretty easily. You can either boil your water to remove it, or you can just let it sit out overnight. Furthermore, you can taste the chlorine in your water; no nasty surprises.
Chloramine is stealthier. You can't taste it or smell it. You won't know it's there unless you read your city's water report. You can only see if it you fill a large, white container with a very large volume of water, and even then it is a very faint blue-green color that you might not even notice. Even worse, chloramine doesn't dissipate as easily. In fact, that's why cities are using it now: it is more likely to stay in your water, so they can use less and be sure that it keeps killing things all the way to your house. However, this means that it won't come out of your brewing water as easily as chlorine would.
Even if you don't use chlorinated brewing water, chlorine can get in your water if you sanitize with bleach and don't rinse adequately. Don't use bleach, just use a proper no rinse brewing sanitizer like Star-San.
If it is soooo hard to remove, why bother? The problem is that yeast produce special compounds called 'phenols'. Phenols are okay in certain amounts and in certain styles. They give Belgians and Hefeweizens much of their character, for example. Phenols can have a spicy, plastic-y, medicinal or clove-like flavor. You definitely don't want too much of that in most beers, *especially* the medicinal, band-aid like flavor.
The problem is that yeast eat chlorine and chloramine and produce 'chlorophenols'. These phenols have extremely low taste thresholds; you can taste them even if they exist in only a few parts per billion. Your beer will have a difficult to define 'sharp' or chemical flavor, and usually a plastic or band-aid aftertaste, like you just stuck your nose into a first-aid kit. If the amount is low enough, you might only be able to taste it when you burp.
My first batch ever used tap water. I used it as top-off to a Mr. Beer Classic American Blonde kit. After two weeks carbing, the first thing I tasted was green apple (acetaldehyde, generally a sign that you used too much simple sugar in your beer and haven't let it age long enough to compensate). The second thing I tasted was medicine.
I started using spring and distilled water, and the problem went away. Starting with my first all-grain, however, I decided to go back to using tap water. I learned that Campden tablets (sodium metabisulphite) could be added to remove chlorine and chloramine at a rate of 1 tablet per 20 gallons, so I used 1/4 to 1/2 tablet per batch. I did the same thing on all future batches: 9, 10, 11, 12, 13 and 14. Then I cracked open batch 8, and had an unpleasant surprise. Yep, medicine burps. My friend swore he couldn't taste it, but there was no sign of the taste in batches 2-7, while batches 8-10 all have it.
Blah blah blah, TLDR, so how do you deal with it? Here are your options, in order of cost:
If you're using chlorine, put all your brewing water in a big container (bottling bucket, mash tun, whatever) and let it sit for a day or two before using it. This does NOT work with chloramine. Cost: None.
Use one campden tablet. Do not bother cutting it in quarters. They say you only need a quarter tablet for 5 gallons, but look how well that turned out for me? There is no harm in using a full tablet, and these things are dirt cheap. Let it sit for at least 5-10 minutes before brewing, or a few hours, or all night like I do. Cost: Pennies.
Buy an activated carbon filter. By all accounts, this thing is the ultimate solution. Activated carbon removes chlorine and chloramine if you run your water through it at a very slow rate, between 2 and 5 minutes per gallon. Start your water flowing at 5 minutes per gallon (experimentation and experience will let you know what that is), set a timer for half an hour or so and come back when it beeps. The guys at MoreBeer use this method on our EBMUD water with good results, and I decided to just go for it instead of worrying constantly about my water. You have to use an activated carbon filter, not a 'brita' or 'pur' filter, and you should NEVER run hot water through it. Cost: $42.95 for the first year, $8.25 for replacement filters every year after that.
Use RO water from your grocery store. I'm sure you've seen the big water vending machine inside or out in front of the grocery store. They usually charge between 30 and 50 cents per gallon and either sell RO (Reverse Osmosis) or 'spring' water. RO water is extremely pure and lacking in minerals, so if you go this route for all grain, make sure you add brewing salts to make up for it. Cost: I use about 8 gallons per batch, and brew a batch per week. If I brew around 45 batches per year and buy my water at 35 cents/gallon at Safeway, that comes out to $126 per year. I also need to buy and store a big plastic jug to carry that water in (two, technically, since they only hold 5 gallons each).
You can also install your own RO system in your kitchen. These are generally expensive, require a lot of setup, and a lot of space for the system itself and the water tank. This is probably cheaper in the long run than bottled water, and is worth it over a carbon filter if your tap water has too many minerals in it or tastes disgusting. I'm not 100% sure if RO removes chloramine.
TLDR: Remove chlorine and chloramine from any water that touches your beer or brewing equipment - that includes sanitation water, just to be safe. Dont' sanitize with bleach. Remove chloramine with 1 campden tablet per 20 gallons or (preferably) an activated carbon filter at a very slow flow rate. I use the MoreBeer filter and a tablet in every batch now. Remove chlorine by letting it sit out a day or two, or any way that works to remove chloramine.
If I can beat this chloramine problem, that's one foe down. All that will be left is temperature control, evaporation rate consistency and fermentation temperature control.