I've always bottled directly after the primary fermentation has finished. What's the point of the secondary fermentation? Can I use a bucket or is it something that really necessitates a glass carboy?
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:
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.
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.
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).
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.
I was told that it helps the clarity of the beer, since the sludge from the primary will be left over in the first fermenter. I use a glass carboy for secondary fermentation.
I also know of many people who bottle right after primary, as you do. If it's working for you, then it's no big deal. Try doing a secondary for a batch and see if you notice a difference.
I agree with Jeff L, that secondary is useful for particular things.
To your question, I only ever do a 'secondary' when adding fruit or extended aging. My beers clear just fine after two weeks in the fermenting vessel. I've never experienced off flavors from that two weeks on the yeast, and I always dry hop in the primary.
For me I can't find any reason to use a second vessel (except for fruit or extended aging, meaning more than a month) so it isn't worth the hassle.
I think Secondary fermentation is a term that came over from wine making. Secondary fermentation in the wine world is where you have the malolactic fermentation happen. This is a bacteria not yeast and it reduces the malic (green apple) acids to lactic acids soften the wine. This usually is done in barrel storage.