All we did was cut the downspout, put a pedestal base in place to elevate the barrel (since it's just like a water tower, in that flow rates correspond to the hydrostatic pressure provided by gravity), made sure it was level, and set the barrel on top, securing it to the house wall with some rope (in the off-chance the base should become unstable).
Of course, we also had to test it, so I put the garden hose in the gutter and turned on the water works. Since I didn't measure the height as accurately as I should have (not easy on uneven ground), I came up a little short of the tip of the downspout; to compensate, I placed a brick on top to prevent water from spilling over the top of the barrel. And, thanks to the barrel's well-fitting screen, which is intended to keep gutter gunk out of the water, Mom probably won't need mosquito dunks.
Given the size of Mom's garden and the fact that they have a sizeable roof that collects plenty of runoff, it should be a wise investment. I'm interested to see how much of an effect it has on their water bill (though specific figures might be hard to quantify). Needless to say, the barrel filled to the brim after just one storm; Mom attached a soaker hose to the bottom, which snakes through her vegetable garden, providing slow 'n' steady moisture to the soil.
Now if you'll excuse me, I have to go perform a rain dance in the backyard.