Its forms

The nominative is actually nothing more than a set of article endings and a few personal pronouns. Those are:

ARTIKEL
der (m)
das (n)
die (f)
die (pl)

PERSONALPRONOMEN
ich = I
du = you (singular)
er = he
es = it
sie = she
wir = we
ihr = you (more than one person)
sie = they
Sie = you (sir/madam)(formal)

You can easily memorize the article endings with help of the keyword: Nordsee with No standing for Nominativ. It is the endings you want to learn as they are used with all kinds of articles, e.g., welch er (m), dieses (n), meine (f), keine (pl)

When do I use it?

The nominative is used for the subject of a sentence. Subjects are those nouns or pronouns that do the action in a sentence.

Der Hund bellt.
The dog barks.

The dog is doing the barking, so it is the subject of that sentence. And you can see that it uses der and not any other of the masculine articles (those would be den, dem or des and you'll get to know them later).
Then there are three other situations after which you will need to use the nominative in addition (!) to the subject: After sein, bleiben and werden (to be, to stay/to remain, to become). Take a look at the following examples:

Der Lehrer ist der Vater (von meinem Freund).
The teacher is the father (of my friend).

Der Hund bleibt der beste Freund (des Menschen).
The dog remains the best friend (of a man).

Der Mann wird der Gatte (meiner Schwester).
The man will become the husband (of my sister).

How do I find the subject?

You can find the subject by asking the question WER? for persons or WAS? for non-persons:

Wer singt? Der Chor. <-- CHOIR IS A GROUP OF PERSONS
Who is singing? The choir.

Was ist kaputt? Das Auto. <-- AUTO IS A NON-PERSON/THING
What is broken? The car.

Do you remember the similarity between wer and der & was and das?