In many languages nouns are marked not only for number—singular, plural, and in some languages dual—but also for case: the role which the noun plays in the sentence. Latin, for instance, has six cases; these are marked with specific suffixes, which are different for different classes ('declensions') of nouns. Here are examples from three different declensions:
1st declension puell(a) = 'girl'
CASE SINGULAR PLURAL TYPICAL USE
Nominative puella puellae subjects
Genitive puellae puellarum possessors
Dative puellae puellis indirect objects
Accusative puellam puellas direct objects
Ablative puella puellis locations
Vocative puella puellae persons spoken to
2nd declension av(us) = 'grandfather'
CASE SINGULAR PLURAL TYPICAL USE
Nominative avus avi subjects
Genitive avi avorum possessors
Dative avo avis indirect objects
Accusative avum avos direct objects
Ablative avo avis locations
Vocative ave avi persons spoken to
3rd declension can(is) = 'dog'
CASE SINGULAR PLURAL TYPICAL USE
Nominative canis canes subjects
Genitive canis canum possessors
Dative cani canibus indirect objects
Accusative canem canes direct objects
Ablative cane canibus locations
Vocative canis canes persons spoken to
Word order doesn't matter, as it does in English; it is the ending which tells you what syntactic role the word plays.
For instance, in the following sentences, donat is a verb meaning (he/she/it) gives, which takes a subject, an indirect object, and a direct object. Puella is marked for the nominative singular, so it's the subject; avis is marked for the dative plural, so it's the indirect object; and canos is marked for the accusative plural, so it's the direct object. These sentences all mean the same thing: The girl gives dogs to the grandfathers.
Donat puella avis canos.
Puella canos avis donat.
Avis canos donat puella.
Canos donat puella avis.
In Modern English, personal pronouns are only a little bit less complicated, typically marked for four or five cases:
CASE SINGULAR PLURAL TYPICAL USE
Subject I we subjects
Object him them direct and indirect objects
Reflexive myself ourselves objects of one's own action
Dependent Genitive my our possessors as determiners
Independent Genitive mine ours possessors as nominals
But with nouns the case system is substantially diminished: there are only two cases:
CASE SINGULAR PLURAL TYPICAL USE
Genitive boy's boys' possessors
Plain boy boys everything else
What the Oxford Grammar tells you is that Modern English nouns do not have distinct object and subject forms. Only the genitive case is specifically marked in nouns, and reflexive case is expressed with the appropriate pronoun; all other syntactic roles are distinguished by word order or the use of prepositions.