It's an important question and a good question. Here's one use of llamar:
At 8:00, I call my cat to come in and eat.
A las 8:00, yo llamo a mi gato a venir a comer.
Sometimes I call my cat "Crazy" because of his high jumps.
A veces yo llamo a mi gato "Locochón" debido a sus grandes saltos.
But that's just a silly nickname. His name is actually Miguelito. To express that, I'll use llamar with a special pronoun tucked before it:
Mi gato se llama Miguelito.
A literal, word for word translation of this would be
My cat calls himself Miguelito.
But a correct translation of this expression would be
My cat's name is Miguelito.
Here's what it looks like, in the other conjugations:
Yo me llamo Samuel.
¿Tú te llamas Samuel?
Nosotros nos llamamos la Familia Pérez.
Vosotros (pardon me if I skip this -- we don't use this in Mexico and I believe it might vary from one country to another, I don't want to steer you wrong -- but someone else is welcome to edit my answer and fix this up)
Mis padres se llaman Conchita y José Manuel.
You could make a word for word, literal translation from English and say:
My parents' names are Daisy and Pablo.
Los nombres de mis padres son Daisy y Pablo.
and people would understand you; but it would sound a bit awkward.