I don't know if this is specifically the answer you are looking for, but I can explain the different case endings for nouns and verb forms.
Let's start off with nouns.
Nouns in the first declension are typically feminine, with the exception of the Latin words for poet (poeta, poetae) and farmer (agricola, agricolae). I do believe that there is one other, but it is slipping my mind at the present moment. Here is a chart of endings for the first declension.
| Singular | Plural |
-----|------------|-----------|
Nom | -a | -ae |
Gen | -ae | -ārum |
Dat | -ae | -īs |
Acc | -am | -ās |
Abl | -ā | -īs |
Nouns in the second declension are masculine and neuter. The case endings are slightly different between the two declensions, so I have noted the neuter variations in parentheses.
| Singular | Plural |
-----|------------|-----------|
Nom | -us (-um) | -ī (-a) |
Gen | -ī | -ōrum |
Dat | -ō | -īs |
Acc | -um | -ōs (-a) |
Abl | -ō | -īs |
Nouns in the third declension can be masculine, feminine, common (can be either masculine or feminine), or neuter. Again, the neuter variations are in parentheses.
| Singular | Plural |
-----|------------|-----------|
Nom | (varies) | -ēs (-a) |
Gen | -is | -um |
Dat | -ī | -ībus |
Acc | -em (1)| -ēs (-a) |
Abl | -e | -ībus |
1 Matches nominative
There is also a 4th declension which contains mostly masculine nouns, and a 5th declension which is all feminine save for the word for "day" which is diēs. I can explain these if you would like, but they are generally less common than the aforementioned three declensions.
You may also be wondering about the cases. I'll give a brief explanation below, as I do not know how much you have investigated about this particular area. Keep in mind that this does not cover all the particular uses for each case, but it should give you a relatively useful foundation.
Nominative (nom): the subject of the sentence (ex: the cat chases the mouse)
Genitive (gen): typically indicates possession (the thief took the queen's crown)
Dative (dat): used for reference and as an indirect object with the words "to" or "for" (ex: she gives to the charity)
Accusative (acc): the direct object of the sentence (ex: the cat chases the mouse)
Ablative (abl): used in various prepositional phrases (ex: the cavalry on the bridge was marching with the king.)
Now for verbs!
There are 4 conjugations of verbs, creatively named first, second, third (of which third-io is a sub-conjugation), and fourth.
You can see examples of the different conjugations in one of my previous answers, as I don't want to make this answer too lengthy.
However, I did not include examples of 3rd-io verbs in that answer, so here are a couple:
capio, capere, cepi, captum = seize, capture
facio, facere, feci, factum = do
To answer your last few questions from a general standpoint, noun declensions and verb conjugations allow you to make all the different forms for words so that they can be used in various purposes.
I hope this explanation helps, let me know if you would like any clarification or additional information!