Russell's preferred example of what he means by a "simple particular" is a sense-datum (an object of sensory experience), but it could apply to any object of awareness that is the sort of thing having properties. Importantly, what we often intend to call "particular" are not really particulars: they are in fact more complex. This includes any object that is not in our field of awareness, like objects in the past beyond our memory and objects that we are not (or never are) aware of, including ordinary objects of common sense. As Russell puts it:
The simplest imaginable facts are those which consist in the possession of a quality by some particular thing. Such facts, say, as "This is white."...What pass for [proper] names [i.e., words for particulars] in language, like "Socrates," "Plato," and so forth...are really abbreviations for descriptions; not only that, but what they describe are not particulars but complicated systems of classes or series. A name, in the narrow logical sense of a word whose meaning is particular, can only be applied to a particular with which the speaker is acquainted, because you cannot name anything you are not acquainted with...It is only when you use "this" quite strictly, to stand for an actual object of sense, that it is really a proper name. (The Philosophy of Logical Atomism, Lecture II, pages 522-525)
So ordinary objects like desks are not particulars, that is, they are not logical atoms. They are logical fictions, that is, series of particulars having common properties:
Now the essential point is this: What is the empirical reason that makes you call a number of appearances, appearances of the same desk? What makes you say on successive occasions, I am seeing the same desk?...There is something given in experience which makes you call it the same desk, and having once grasped that fact, you can go on and say, it is that something (whatever it is) that makes you call it the same desk which shall be defined as constituting the same desk...In that way the desk is reduced to being a logical fiction... (The Philosophy of Logical Atomism, Lecture VIII, pages 369-370)
So Russell thinks that many ordinary objects are series of particulars having some common experienced property (like the same color, shape, odor, and so on). The real particulars are the relatively short-lived objects of our immediate experiences.