Although I have already tried to show why the traditional definition of a noun (person, place, or thing) is inadequate, now that we have come to define what nouns are, I am going to start with that definition anyway. Am I contradicting myself? Not really. Nouns do refer to people, places, and things, but that doesn't exhaust the extent of their reference. People, places, and things are prototypical nouns. If we're studying a new language, the category that we will call "noun" in that language will be the one that includes these core objects.[1] We will start with these core nouns, observe the patterns that they exhibit, and then use those patterns as a structural test for other words whose category membership may be less clear.

Let's begin with a few examples of such core nouns: teacher, house, car.

All of these words use the same suffixes. They change form to distinguish singular from plural by adding -s:

One teacher, two teachers
One house, two houses
One car, two cars

They also take a different suffix that is traditionally called the possessive ('s for singular nouns, s' for plural ones) although for reasons we will come to later, it's more accurately called the genitive.

the teacher's credential
the house's roof
the car's engine

Nouns can also be formed from preexisting verbs, adjectives, other nouns by adding certain suffixes, e.g., -ment, -tion, -hood, etc. So the presence of such a suffix is good evidence that the word you are looking at is a noun. (See the section on morphology for more information on this process.)

These morphological tests work for a wide variety of nouns, but not all. For example, there are some nouns that form the plural irregularly (e.g., mouse/mice), or show no difference in form at all (e.g., sheep, deer, etc.). Nevertheless, we still want to assign these words to the same category because in other respects they behave just like the more regular words.

Another set of tests looks at the context in which a word can appear in phrases or sentences. As was indicated above, nouns can appear in sentence structures such as the frame in (4), repeated here for convenience:

(4) She has no _____.

Nouns can also appear as the subjects of sentences:

(11) Deprivation is growing among the unemployed.

Nouns also follow certain function words known as determiners (see below), such as the, a(n), my, that, etc. Thus we can say the enrollment, but not *the enroll.

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Notes

[1] Although the claim is not entirely uncontroversial, most linguists believe that every human language distinguishes at a minimum between nouns and verbs.