The term "secondary fermentation" is misleading since the purpose isn't to continue fermentation. A secondary stage can be used for any combination of things:

1. Clarification: racking to secondary gets the beer off the yeast cake and allows more particulates to fall out of suspension. This is often the only reason I use a secondary stage; I like clear beer.

2. Dry hopping: when making an IPA or other beer where dry hopping is involved, it is almost always done in secondary.

3. Flavor additions: adding things like fruit, spices, coffee, chocolate, etc. is almost always done in the secondary stage.

4. Aging: if aging a beer for a long time (months to years), transferring it to a secondary removes it from the yeast cake and frees up your primary fermenter for another beer.

I wouldn't recommend a bucket since you want to minimize head space to minimize exposure to oxygen. A bucket often provides too much beer-to-air contact, especially if aging a beer for an extended period of time. Some will argue that glass carboys are necessary, but I use plastic carboys and haven't had any problems.

The term "secondary fermentation" is misleading since the purpose isn't to continue fermentation. A secondary stage can be used for any combination of things:

1. Clarification: racking to secondary gets the beer off the yeast cake and allows more particulates to fall out of suspension. This is often the only reason I use a secondary stage; I like clear beer.

2. Dry hopping: when making an IPA or other beer where dry hopping is involved, it is almost always done in secondary. If you tried to dry-hop during primary fermentation, much of the hop aroma would be lost through the airlock with the CO2 that escapes during fermentation.

3. Flavor additions: adding things like fruit, spices, coffee, chocolate, etc. is almost always done in the secondary stage. Often this is for the same reason as dry hopping (to ensure the aroma and flavors are retained).

4. Aging: if aging a beer for a long time (months to years), transferring it to a secondary removes it from the yeast cake and frees up your primary fermenter for another beer. In cases where a beer is going to be aged for 6 months or more, removing it from the yeast cake also prevents autolysis.

I wouldn't recommend a bucket since you want to minimize head space to minimize exposure to oxygen. A bucket often provides too much beer-to-air contact, especially if aging a beer for an extended period of time. Some will argue that glass carboys are necessary, but I use plastic carboys and haven't had any problems.