A Champagne bucket is used to quickly cool Champagne down to drinking temperature, 45 to 50 degrees Fahrenheit, or 7 to 10 degrees Celsius, and keep it at optimal drinking temperature. It's really just a special ice bucket, and an ice bucket of the proper size works just as well.
Champagne shouldn't be stored long term in the fridge. It can be cooled not so quickly in the fridge, taking about 4 to 5 hours, or much more quickly in a Champagne bucket. The typical technique is to put the bottle of Champagne in the bucket, add ice to the rim, and then half fill it with water. 20 or 25 brief minutes later the Champagne is ready to drink. Personally I fill the bucket all the way with water, not just half way, and it does cool faster.
The bottle can be placed back in the bucket to keep it cool, but often it will get much too cold and simply not taste as well. Now temperature is somewhat of a personal temperature, but "ice cold," 32 degrees Fahrenheit or 0 degrees Celsius, is too cold. I will normally but the bottle back in the bucket occasionally for 5 minutes or so.
Different Champagnes will drink better at different temperatures, although this isn't worth stressing out over. In general, older Champagne are better slightly warmer. A bottle of 66 Dom Perignon for example, was pretty good shortly out of the refrigerator, but only truly blossomed and was incredible after about 45 minutes warming up.
More Champagne is drank too cold than too warm. For example, right out of the fridge you should never put it in the bucket as it is a bit colder than ideal (although I usually pop the cork and pour immediately, letting it warm up and open up gracefully in the glass). Of course some cheap sparkling wine should be served extremely cold as it doesn't taste good and you want to hide the flavors (note: there are decent cheap sparkling wines too, and my tastes usually direct me towards Cava, a sparkling wine from Spain).
The Godinger Crystal Winchester Champagne Bucket to the left is elegant, functional, and not outrageously priced (I've seen $4000 buckets before!). It is 11.2" by 12.1" by 21.4" in size, and the hob nail and pineapple cuts in the crystal look great.
A less expensive and more utilitarian model is the Stainless Steel Champagne Bucket to the right. It is made of reflective 18/10 stainless steel, dishwasher safe, and 7-1/2" tall and 7-1/2" wide. Two metal rings, one on each side, make it easy to carry.
So what if you need to chill Champagne quickly and have no Champagne bucket? Well, you could use any old bucket. I tend to use a large pot, or if traveling, the bathroom sink. This works especially well in hotels, as they almost always have ice machines handy!