The bucket object implements a bucket brigade device (in the electronics world, this is sometimes referred to as a discrete time-analogue delay line). As the examples below suggest, it's useful for single-sample delay lines, calculating averages, computing velocity and acceleration, and permutating lists by rotating them.
In the Max tutorial on abstractions, you can find a classic example of why the bucket object is so useful - this Max tutorial demonstrates its use as a single sample delay for number values useful for constructing drawing commands.
Lots of people use the bucket object for shifting data in series, it's computationally useful for calculating first and second derivatives for streams of data. The terms "first and second derivative" are math-speak for keeping track of:
Here's a patch that does that:
I also tend to use the object quite a bit in situations where I desire a running average over some finite set of events (if you want a running average for an stream or unspecified set of values, you'd use the mean object). Here's an example that calculates the average of the last 8 floating point input values:
The bucket object is a little unusual in that you don't need to worry about using arguments to the object in order to have it work with floating point values. In fact, bucket is so useful that people often ask whether or not there's a Max object that is similar but will work with symbols or lists. While there isn't a specific object, here's a Max patch that does exactly that (and it also helpfully outputs stuff in standard right-to-left order, too):
One way to think of the bucket object is a as an object that adds new input values and rotates the list by one position. While you can rotate a list forward or backward using the r2l and l2r messages to the bucket object, performing arbitrary list rotations can be done using the zl object and specifying the rot option:
The zl object (in its zl rot form). Useful for performing rotations of lists where the list offset is an odd number greater than 1.
The cycle object
(please list things that you believe to be errors or omissions from the existing refpage)