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Bertrand Russell in Principles of Mathematics defines a term as "Whatever may be an object of thought, or may occur in any true or false proposition or can be counted as one." Can someone elaborate on this and give an example of something which is not a term?
Russell continues later on by saying that in the sentence, "Socrates is human", we have only one term, Socrates, a verb, and a predicate. Why is human not considered a term?
The answer already given by user3451767 is not in my view correct. Logical atomism characterizes Russell's work from roughly 1910-1925. POM was published in 1903. There are many differences between his views during these periods.
The short answer to your question:
Russell's terminology is confusing because he has two uses of the word "term" in POM:
-A term "ontologically" speaking is anything that can be a subject of a proposition. So the concept /human/ is really a term because it can occur as a subject in a proposition (e.g. /humanity characterizes Socrates/).
-But Russell also speaks about terms of the proposition, and by this he means the way a term occurs in a proposition. So in the proposition /humanity characterizes Socrates/ the term /humanity/ occurs as the term of the proposition, and /characterizes/ occurs as the relating relation. But in the proposition /Socrates is human/ the term /humanity/ does not occur as a term of the proposition, but as the relation of the proposition.
Nicholas Griffin (1980 pp.120-122) suggests that because this ambiguous terminology is misleading terms of the proposition (in contrast to terms tout court) should be simply called the subjects of the proposition.
So there is no contradiction between your examples: In the first example Russell is using the word "term" in the first sense, and in the second example he is using it in the second sense (terms as subjects of the proposition in question).
The longer explanation:
In the period when POM was written Russell had a very simplistic view about language (and thought), according to which almost every word in a sentence has a meaning and what it means is a term (CCBR p.20). So in a sense his ontology derives from this kind of simple view of semantics, but really he was not interested in language here:
Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary.[...] A man, a moment, a number, a class, a relation, a chimera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false. (POM p.43)
Terms and propositions are neither linguistic nor psychological, but objective constituents of the world. Anything that can be counted as one or made the subject of a proposition is a term. (CCBR p.20 & p.115) Russell also has a habit of using words like "verb", "adjective" and even "variable" in a non-linguistic sense, as referring to constituents of the non-linguistic propositions.(Griffin 1980, n4)
In short his theory of propositions is the following (Griffin 1980, p.119): All terms have being (Zeus), in some cases they have existence (the moon). Terms are divided into things and concepts. So concepts are also terms, but concepts have the ability to occur in propositions as the relation of a proposition, and not only as one of the subjects of a proposition. And when terms (things or concepts) combine with concepts they form new complex terms, these are propositions. (So propositions are also terms!) Propositions are either true or false, and these are unanalyzable properties of propositions (so for example truth is not based on a correspondence relation for Russell in POM).
So for Russell in POM both
(1) /Plato admires Socrates/
(2) /Admires admires Socrates/
are proposition (sometimes in the literature slashes are used instead of quotation marks to emphasize that propositions here are not linguistic entities). (1) is true, and (2) is false (so he does not impose type restrictions). (Griffin 1980: 121)
What does it mean to say that concepts but not things have the ability to occur as a relation in a proposition? A relation is what gives the proposition its unity according to Russell. But how exactly and what this unity consists of Russell was never able to specify (this is the so called problem of the unity of propositions). (CCBR: pp.116-117)
This partly Meinongian theory is very strange, no doubt. But its purpose is clear: Russell needed a theory of propositions for his logic, since logic was for him (in POM) the science of propositions.
Now to your first question: give examples of things that are not terms. According to Griffin (1980, p. 119) when things are combined with other things the result is not a term but an object, objects (unlike terms) need not be unities. Examples of such objects are for Russell classes-as-many and the denotations of quantifiers (in POM Russell accepted the theory of the so called denoting concepts).
Notice that Russell also called terms objects, so apparently object is a more encompassing category. At §58 n1 (POM) he finds this problematic:
I shall use the word object in a wider sense than term, to cover both singular and plural, and also cases of ambiguity, such as "a man." The fact that a word can be framed with a wider meaning than term raises grave logical problems.
Russell returns to this issue at §348 (p.366).
Griffin, Nicholas (1980). Russell on the nature of logic (1903–1913). Synthese 45 (1):117-188.
CCBR: The Cambridge Companion to Bertrand Russell
Some quotations from Principles of Mathematics (1903), in order to give more context.
§46. Of the parts of speech, three are specially important: substantives, adjectives and verbs.
§47. Philosophy is familiar with a certain set of distinctions, all more or less equivalent: I mean, the distinctions of subject and predicate, substance and attribute, substantive and adjective, [...]. Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words unit, individual and entity.
Here Russell uses the words "term" and "propiosition", not always in a consistent manner, to refer to "parts of the world" instead of "parts of speech": propositions are not linguistic entities [see §46 : "the sentence expressing the proposition"] :
§51. Words all have meaning, in the simple sense that they are symbols which stand for something other than themselves. But a proposition, unless it happens to be linguistic, does not itself contain words: it contains the entities indicated by words.
Thus, a term is not a word but an entity denoted by a word (an individual one), being it a particular or a "general" one.
§48. Among terms, it is possible to distinguish two kinds, which I shall call respectively things and concepts. The former are the terms indicated by proper names, the latter those indicated by all other words. [...] Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs. The former kind will often be called predicates or classconcepts; the latter are always or almost always relations.
Points, instants, bits of matter, particular states of mind, and particular existents generally, are things in the above sense, and so are many terms which do not exist, for example, the points in a non-Euclidean space and the pseudo-existents of a novel. All classes, it would seem, as numbers, men, spaces, etc., when taken as single terms, are things [...];
§49. The above argument proves that we were right in saying that terms embrace everything that can occur in a proposition, with the possible exception of complexes of terms of the kind denoted by any and cognate words.
Here we can say that Russell shows its relations with Frege.
See this post for some comment on Frege's views.
It seems that Russell's use of "term" is an attempt to introduce a "common category" encompassing the Frege dualism between object and concept (or function).
In particular, terms denoted by adjectives or verbs (i.e.concepts and relations) have not the "unsaturated" (and somewhat inconvenient) nature asserted by Frege.
At the same time, it seems that Russell does not agree with Frege radical departure from natural language - to be supplanted into scientific relam - by the "perfect" concept-script :
§46. In the present chapter, certain questions are to be discussed belonging to what may be called philosophical grammar. The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. Although a grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is primâ facie evidence of the other, and may often be most usefully employed as a source of discovery. Moreover, it must be admitted, I think, that every word occurring in a sentence must have some meaning: a perfectly meaningless sound could not be employed in the more or less fixed way in which language employs words. The correctness of our philosophical analysis of a proposition may therefore be usefully checked by the exercise of assigning the meaning of each word in the sentence expressing the proposition. On the whole, grammar seems to me to bring us much nearer to a correct logic than the current opinions of philosophers; and in what follows, grammar, though not our master, will yet be taken as our guide.
Note
As rightly commented above, Russell is not always consistent; sometimes he use the "predicate" to refer to a linguistic items, but sometimes he uses it to refer to constituents of propositions:
§48. Thus we shall say that “Socrates is human” is a proposition having only one term; of the remaining components of the proposition, one is the verb, the other is a predicate.
If here we read "proposition" as referring to a non-linguistic entity, we have to read "predicate" and "verb" as referring to non-linguistic entities as well.
Added
It may help to try to find the "backgroud" of Russell ("strange") terminology.
We can refer to the widespread logical treatise of the Oxonian Richard Whately : The Elements of Logic whose publication in 1826 gave a great impetus to the study of logic throughout Britain and the United States of America at the beginning of XIX Century.
See page 55:
Language affords the signs by which these operations of the mind are expressed and communicated. An act of apprehension expressed in language, is called a term ; an act of judgment, a proposition; an act of reasoning, an argument; (which, when regularly expressed, is a syllogism;) as e.g.
"Every dispensation of Providence is beneficial ;
Afflictions are dispensations of Providence,
Therefore they are beneficial:"
is a Syllogism ; (the act of reasoning being indicated by the word "therefore,") it consists of three propositions, each of which has (necessarily) two terms, as "beneficial," "dispensations of Providence," etc.
A century later, Russell - folowing his move from idealism to realism - has shifted the basic "correspondance" between language and mind to that between language and "reality".
Thus, he had adapted the same terminolgy : term and proposition to refer no more to "acts of mind" (apprehension and judgment) but to "parts of the world", i.e. real entities: "objects" and "facts".
