You've already got plenty of good answers - but here is my take.
Danish adjectives are inflected according to number, noun gender and definiteness.
Generally Danish adjectives come in 3 forms (basic form, t-form and e-form).
Ny, nyt, nye (new)
Stor, stort, store (big)
Høj, højt, høje (tall/high)
(This is the basic pattern, but there are a few irregularities with some adjectives)
1) Now first you decide weather the adjective is:
(a) plural
(b) preceded by a word calling for the definite form:
(definite or demonstrative articles (den, det, de, denne, dette, disse (den her, det her, de her, disse her....), possessive pronouns (min, din, vores, deres, jeres) or the possesive form of a proper noun (Peters, kattens, Maries, hundens)
..... if either (a) or (b) you should use the definite form, the e-form
2) If the adjective is neither plural nor preceded by anything calling for the definite form, the only thing that matter is the gender of the following noun. If the noun is common gender, you use the basic form. If it is neuter you use the t-form.
.......................
I know this sounds really complicated, but it isn't really once you've gotten the hang of it:
1) Adjectives have 3 forms and their formation is usually regular
2) definite or plural -> e-form
3) not definite or plural -> basic form or t-form (depending on the gender of the following noun)