That is true, however the one used depends on the grammatical gender of the word. If the word is neuter (a t-word) then you use "mit", if the word is common gender (an n-word) then you use "min", if a word is plural you use "mine". Grammatical gender in Danish can be very arbitrary, some rules of thumb to exist, but there are many many exceptions and gender must be learnt with the noun.
A word that is neuter gender and so uses "et" to mean "a(n)", "det" to mean that (and any other determiner that changes based on grammatical gender) and adjectives end in "-t" (other than adjectives with certain endings, such as "-sk" and "-e"). This post will hopefully help you to determine what gender a word is.