This is the 194th most frequent Danish word.
dig
The Danish word "dig" means "you" (objective form).
The word 'dig' here is used as a direct object for the first person 'jeg', hence 'dig' translates to 'you'.
Jeg vil gerne se dig.
I would like to see you.
Here, 'dig' is the indirect object of the sentence, indicating the recipient of the gift, translating to 'you'.
Jeg har en gave til dig.
I have a gift for you.
In this question, 'dig' is again the direct object, referring to the person being spoken to, equivalent to 'you' in English.