This text was written by Claudio Marcos and Paulo Marcos, the authors of this website, and our aim here is to explain Sanskrit declension.
In Sanskrit, there are three grammatical genders: masculine, feminine and neuter, and three grammatical numbers: singular, dual and plural. Besides that, there are eight grammatical cases: Nominative, Vocative, Accusative, Instrumental, Dative, Ablative, Genitive and Locative. In Sanskrit, any noun, adjective, numeral or pronoun needs to be properly declined before it can be placed on a sentence. So, declension is the process of inflecting a noun, adjective, numeral or pronoun according to gender, number and case. By "inflecting", we mean changing the ending of the word.
To explain further: every noun in Sanskrit has a crude (uninflected) form, called prātipadika. This crude form is the form in which nouns and adjectives are generally found in a dictionary. For example, aśva is a masculine noun that means "horse". Aśva is the uninflected (crude/prātipadika) form of this noun. In general, this crude form can't be used "as is" in a sentence; as we already said above, it needs to be properly declined in gender (in this case, masculine), number and case.
So, for example, aśva declined in the Nominative case and singular number is aśvaḥ, and in the Dative case and singular number is aśvāya. Notice that, in the first case, a Visarga ("ḥ") was added to the crude form of the noun, and, in the second case, the final letter "a" was dropped and the ending "āya" was added.
But what is grammatical case? Native English speakers are familiar with the meaning of gender and number, but not of case. There is no such concept in modern English, at least as far as we know. Probably the best way to explain what case is all about is to explain the meaning of the 8 cases of Sanskrit.
For a very good explanation of the meaning of each case (and of Sanskrit declension in general), we recommend that you read the first page about declension in the Sanskrit & Sánscrito website. But we will explain it here as well.
You should keep in mind that what we will explain here is just the general meaning of each grammatical case (the page from Sanskrit & Sánscrito that we mentioned just above gives you additional meanings and examples).
Nominative: the Nominative case indicates the subject of a sentence. The subject is simply the term that the rest of the sentence says something about. If the sentence has an action verb (a verb denoting action, like "to walk" or "to eat"), then the subject can be seen as the agent of the action described by the verb of the sentence. For example, in the sentence "The house is big", "the house" is the subject, and, in "The monkey eats the banana", "the monkey" is the subject. So, when a noun exerts the role of subject in a sentence, it must be declined in the Nominative case.
For example, a simple phrase like "The horse drinks water" could be translated into Sanskrit as:
अश्वः पिबति जलम् - Aśvaḥ pibati jalam.
Notice that the Nominative form of aśva (aśvaḥ) is being used in the above example.
Vocative: The Vocative case indicates direct address, or, in other words, it is used when you address another person explicitly in a sentence. For example, in the phrase "Come here, John", "John" is a vocative expression.
For an example in Sanskrit, there is the following sentence in the Gurugītā:
त्वं कृपां कुरु मे स्वामिन् - Tvaṁ kṛpāṁ kuru me svāmin.
The above phrase means: "Have mercy on me, master". The word for "master" here is svāmin, and svāmin is the Vocative singular of the noun svāmin, which means "master".
Instrumental: The Instrumental case generally indicates situations where, in English, the prepositions "with" or "by means of" would be used. For example, it can indicate the instrument of an action (e.g.: "I cracked the nut with a sledgehammer") or it can indicate company (e.g.: "I had dinner with my friend."). As a Sanskrit example of instrument of an action:
धनुषा बाणान् मुमोच - Dhanuṣā bāṇān mumoca.
The above phrase means "(S)he released arrows with the bow". Dhanuṣā means "with the bow" or "by means of the bow", and is the Instrumental singular form of the neuter noun dhanus, which means "bow".
Accusative: The Accusative case usually indicates the direct object of a sentence. In a sentence with an action verb, the direct object is simply the receiver of the action described by the verb. In the example already given above for the Nominative, "The horse drinks water", "water" is the direct object. In the Sanskrit translation given above for this phrase, jalam is the Accusative singular form of the neuter noun "jala", which means "water".
Dative: the Dative case is generally used where in English prepositions like "to", "for", "for the sake of" would be used. For example, the well known mantra:
ॐ नमः शिवअय - Om̐ namaḥ śivāya.
In the above phrase, "namaḥ śivāya" means "salutation to Śiva". Śivāya means "to Śiva" and is the Dative singular form of Śiva.
Ablative: The Ablative case is used where in English prepositions like "from", "on account of", "since", "because of", etc. would be used. For example, शिवात् śivāt, which is the Locative singular of Śiva, means "from Śiva" or "because of Śiva", depending on the context.
Genitive: the Genitive case indicates possession/belonging. It is often used in cases where, in English, the preposition "of" would be used. For example, रामस्य rāmasya is the Genitive singular of Rāma. Therefore, it means "of Rāma" or "Rāma's".
Locative: the Locative case generally indicates location (for example, situations where, in English, the prepositions, "in", "on", "at", etc. would be used). For example, ग्रामे grāme is the Locative form of grāma (masculine noun), which means "village". Therefore, grāme means "in the village".
Note about Sandhi rules: Since declension is about combining a stem with an ending, Sandhi rules come up very often in the process. Sandhi rules are the rules by which Sanskrit words (or prefixes, endings, etc.) change when they are placed together or combined. If you are not familiar with Sandhi rules, we recommend that you read the documents about Sandhi (Combination) in the Sanskrit & Sánscrito website before attempting to study declension (or any other topic in Sanskrit grammar).
This complete list of Sandhi rules is very useful if you want to look up a particular rule.