I think you know what plural is, English nouns also can be plural and singular (dogs vs. dog), so only 'genitive' is left to explain.

Russian nouns have several case forms (6 of them, according to the most common classification). Each form is used to convey some grammatical meaning.

Nominative is the most basic form, it's used for the grammatical subject of the sentence (соба́ка сиди́т 'the dog is sitting') and for the predicate in «X is Y» sentences (in «соба́ка зла́я» 'dog is angry', both собака and злая are nominative).

Genitive is often translated with the 'of' proposition, or it's sometimes translated with the English possessive case: соба́ка сестры́ sister's dog (сестры is genitive). However, it has a number of other uses: notably, after «нет» is is used to express absence.

Dative corresponds to the English preposition 'for': соба́ке 'to dog'.

Listing all the cases and their functions here might not be useful, as you'll learn all of them as you progress through the course. To get a list of all the case forms, you could use Wiktionary: https://en.wiktionary.org/wiki/%D1%81%D0%BE%D0%B1%D0%B0%D0%BA%D0%B0#Declension

And what is a "genitive plural form"?

Here's an outline:

:) It's from an old Russian grammar.

Actually, it's not quite so bad as it appears at first sight.