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So I recently learned the first declension nouns concept and my list of new vocabularies to learn suddenly changed.
Words were usually presented as:
Now, they indicate the following:
I assume the -ae is nominative plural? Of what use is this indication? I feel like there's something I'm missing here.
The second form is the genitive singular.
The above answers address the substance of your question, but I wanted to add as a supplementary answer something that doesn't appear to be explicitly stated elsewhere: The genitive singular is the only form that is unique and uniform for each declension.
This means that if I know the genitive singular, I know the declension of a word. This is why dictionaries and learning textbooks include this second form.
Here is a chart of the genitive singular for each declension:
Combined with knowledge of the gender of the noun and the nominative singular (necessary for 3rd declension nouns), I can form all the inflections of (almost) any Latin noun.
Here are two examples where this second entry is necessary:
It's the singular genitive-case form. The use is that if you know the nominative singular form, genitive singular form, and the gender of a regularly inflected Latin noun, you can predict all of its other forms. These are the "principal parts" of a regular Latin noun. (Some nouns have irregularities in their inflection that require more description, like the irregular first-declension noun dea "goddess" which has the ablative plural deābus.)
Being given the genitive may seem a bit useless for nouns that end in "a" in the nominative singular, since nearly all of these are regularly inflected according to the first declension. But there are certain Greek loanwords that end in "-a" in the nominative singular and belong to other declension classes and/or genders, such as problēma, problēmatis n. and a number of other neuter third-declension nouns from Greek ending in -ma, matis. (See this related question that mentions a story about declining one of these nouns incorrectly: A story of a king who wanted to simplify Latin grammar)
And knowledge of the genitive form is frequently necessary to be able to inflect nouns of the third declension, where nominative singular forms often don't contain enough information to determine the other forms. For example, the nominative plural of rēmex is rēmigēs, while the nominative plural of caudex is caudicēs.
The first form, e.g., cena, is the lemma or lexical form. This is the form you use when looking up words in a dictionary or lexicon.
The second form is the lemma declined in the genitive case, singular number. In this instance, it is cenae.
Robert J. Henle wrote,1
All Latin nouns are divided into five main groups called declensions, and in these groups we have a model which shows us the proper endings to use. All we have to find about a new noun is this: What declension does it belong to? This we can tell from the Latin case called the GENITIVE. The GENITIVE SINGULAR always tells to what declension a noun belongs.
Furthermore, the genitive singular declension is important because by removing the case ending, we can determine the word’s stem. For example, the case ending for the first declension noun in the genitive case, singular number is -ae. By removing that, we can determine the stem, which is cen-.
Henle, Robert J. Latin: First Year. Chicago: Loyola Press, 1958.
1 p. 5
