Imagine representing one state of a system as a factorial representation, say with one channel for shape and one for position. If there is an obvious best interpretation for the input in those terms--such as single object with clear shape and position--then the factorial representation is a great way to unambiguously represent it, as in the picture below:
But a common problem with factorial representations appears if there is more than one object present. In that case, there will be more than one value of each channel to represent (e.g. multiple shapes and multiple positions). And then it is no longer clear which values should be grouped together (e.g. which shape goes with which position), as below:
This ambiguity is called the "binding problem," and its solution involves grouping the appropriate values together, like {shape1, position1} and {shape2, position2}. In the image below, those values have been explicitly labelled by color; in a software system, they could be grouped into a common structure or given a common index, or in a neural system they could have a common time-varying signal superimposed on them.
A similar problem can occur even if the system is only supposed to represent a single object, but the data is ambiguous. In many implementations this ambiguity is dealt with by temporarily letting the system represent multiple, simultaneous beliefs or hypotheses about the uncertain data (at least until more data, from elsewhere, can help choose which hypothesis is correct). Each of those beliefs is called a "particle"; keeping track of multiple particles at once has proved very fruitful for the automatic analysis of confusing, real-world data.
But representing multiple beliefs about data is exactly like representing multple objects (except that the beliefs are mutually exclusive, while the objects can co-exist together). Other than insisting that the belief-probabilities sum to at most 1.0 total, the two kinds of representation are isomorphic: both face the same binding problem of figuring out which factorial values go together.
The bad news is that representing multiple objects, or multiple beliefs about data, both face the same difficult problem. The good news is that whatever algorithm or circuit solves one problem can almost effortlessly solve the other, so that both belief-particles and multiple-objects representations can coexist side by side, perhaps indistinguishalby, in the same system.