This is an extremely common topic. I recommend using the search to find previous discussions on the topic.
"Secondary fermentation" is a misnomer for beer brewing in almost all cases. It's really a "bright tank" or aging tank, since there is very little fermentation going on. The idea is to take the beer off the trub/yeast cake so it can age without its affecting the flavor.
However, in a home brewery, there's not much effect of leaving the beer on the trub, so that's not a terribly good reason to do it. The beer can age just as well sitting atop its trub, and skipping a racking reduces the chances for contamination or oxidation, so many people don't bother with a "secondary."
There are reasons to do it, though. Some people prefer to rack onto additions like fruit or for dry hopping. If you have a larger primary fermentor, you may need to rack off to free it up for another batch. Depending on the amount of trub, it may help you keep that out of your bottles by leaving some behind when racking to the secondary, and then leaving a bit more behind when going to the bottling bucket. (In general, this won't make a difference, but if there's a whole lot or if it's loose, having less to stir up can help)
If you're doing a secondary stage, the best option is a carboy because of its smaller neck. This means less surface area exposed to oxygen. After primary fermentation, you no longer have a lot of CO2 coming out to push the O2 out of the airlock, so you really want to have a narrow-necked container that exactly fits the quantity of beer. But it certainly *can* be done in a bucket, and in some cases (such as fruit steeping) may be necessary.