der Vater
father
The male parent of a child.
Mein Vater bringt mir bei, wie man einen Reifen wechselt.
My father is teaching me how to change a tire.

father
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father
The male parent of a child.
Mein Vater bringt mir bei, wie man einen Reifen wechselt.
My father is teaching me how to change a tire.

When talking about your own father or addressing him directly, the article can often be omitted. This is common in family and spoken language.
Hast du Vater schon gefragt, ob er uns hilft?
Have you already asked Father whether he will help us?
In the figurative meaning 'Begründer' (founder), 'Vater' (father) is often combined with a genitive or a 'von' (of) construction. This indicates what the person is the originator of.
Er gilt als der geistige Vater dieser Reform.
He is considered the intellectual father of this reform.
The form of address 'Vater' (Father) for God is specific to the Christian faith. A very well-known fixed expression is 'Vater unser' (Our Father), the beginning of the most important Christian prayer.
Im Gottesdienst beten die Gläubigen zum himmlischen Vater.
During the church service, the believers pray to the heavenly Father.
mother / nut
'die Mutter' and 'der Vater' both refer to a child's parents. The difference is gender: the mother is the female parent, and the father is the male parent. The words can be confusing for learners because they appear in the same family context and have different articles: 'die Mutter', but 'der Vater'. In everyday situations, both are often used together, for example when talking about family, upbringing, or parents.

Der Vater bringt seine Tochter morgens zur Schule.
The father takes his daughter to school in the morning.
Meine Mutter ruft mich jeden Sonntag an.
My mother calls me every Sunday.