Storing brown sugar in a tightly sealed container (such as tupperware, rubbermaid, etc.) is the best method.
Once it dries out however it can be re-moisturized by placing a piece of apple or bread with it inside a tightly sealed container.
After a day or two the brown sugar will soften and the bread will dry up or the apple will shrivel.
This is due to the hygroscopic nature(ability to absorb moisture from the surrounding atmosphere)of brown sugar.
You'll see "brown sugar keepers" in gourmet and cookware shops that are round decorative unglazed ceramic or terra cotta disks.
The concept is to soak them in water for an hour or two and then dry and place in the brown sugar.
Save your money and use a piece of bread or apple when the brown sugar is drying up.
You can also soften it for immediate use by microwaving for a little bit.
This should only be used for immediate use because after the sugar cools it will have lost more moisture (due to the fact that microwaves are heating the moisture in food).
cooking.stackexchange.com/users/426/Darin Sehnert
I use a ziplock freezer bag and get as much air out as I can, and generally it keeps pretty well, 6 months or more.
When it does get hard, slice a piece of apple and put it in the bag with the sugar for a couple of days.
There are also ceramic disks you can buy that you can put in the bag with the sugar that will keep it soft for much longer, but I've never used one myself so I can't vouch for it.
cooking.stackexchange.com/users/178/Tim Sullivan
i happen to have one of those seal-a-meal vacuum sealers, and i use one of their container options and store the opened bag in that.
i have heard that the apple/bread slice trick works, too, so would give that a shot instead of buying some special thing.
cooking.stackexchange.com/users/1415/franko
I've found the best way to prevent it from happening is by keep a terra cotta stone in my bag/tupperware holding the brown sugar.
Something like this or this
(the latter says it keeps for 3) works great for me.
You wet it and leave it in there
and it keeps the brown sugar from drying out.
You'll have to re-soak it every so often, but it's not weekly or even monthly in my house, and I open the brown sugar so often
it's easy to remember to do it when I notice the stone is dry.
cooking.stackexchange.com/users/1236/stephennmcdonald
For a quick fix, you can re-soften it easily by putting a slice of apple in with it for a bit.
Don't forget about it though, or the apple will go mouldy and ruin the sugar
(I speak from experience on this one!).
A better option with no such downside is a "brown sugar disc".
It's a piece of clay that you soak in water for 15 minutes, then put in with the sugar.
It will help keep the sugar soft, without getting it too moist.
You can usually find these easily in kitchen shops (or even grocery stores, sometimes).
Every few months you may need to re-soak the disc.
cooking.stackexchange.com/users/4504/Allison
My mum used to store sugar in sealable pots with a layer of dried rice at the bottom.
The idea was the rice would act as a desiccant.
These days you could watch out for one of those little packets that seem to often come with electronic products like hard drives etc.
Those little sachets are full of silica granules which are highly absorbent.
You might toss one of those in to the top of the canister.
Don't worry, they are totally harmless to food and may even help with odour contamination as well.
cooking.stackexchange.com/users/335/octonion
I have found those Lock n Lock containers make THE BEST storage for brown sugar.
I have had brown sugar out of the bag stored in one for over a year without it hardening.
No clay discs, no bread pieces, no apple slices....
nothing.
Awesome containers for this kind of storage.
cooking.stackexchange.com/users/24097/Marcy Timmreck
I put a clothes pin on my opened bags of brown sugar and keep them in a dark pantry at room temperature.  Since I don't use it very often, it becomes hardened.  My work-around is to use a grater to grate it but it would be useful to avoid the problem in the first place.  How should I store it to prevent hardening? Do I put it in the freezer? If so, does it need to come to room temperature before using it?
How to store brown sugar without it becoming hard?
cooking.stackexchange.com
<storage-method><brown-sugar>
cooking.stackexchange.com/questions/3935
cooking.stackexchange.com/users/1229/kiamlaluno
Placing it in a sealed container or freezer bag with as little air as possible will help to prevent this from happening. Storing with rice or a terracotta stone will keep it soft for longer, and adding a slice of bread or apple will help to soften hardened sugar. Keep the sugar in a sealed bag or container. If it still hardens, put a slice of apple in with it until it shrivels up. The sugar should have loosened up.
Keep the sugar in a sealed container or ziplock bag or vacuum seal the plastic bag. Try putting a slice of apple or bread in the bag to absorb moisture. You can also buy special ceramic disks for this purpose.