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