"Please open that door."
They are different verbs
開ける akeru (to open - transitive) is an ichidan verb, so to make the -te form the final る is dropped and て is added to make 開けて akete
開くaku (to open - intransitive) is a godan verb ending in くso the く is dropped and いて is added to make 開いて aite
and then there's also hiraku 開く which conjugates the same as aku and is both used for transitive and intransitive with a slightly different nuance
When to use each 'to open' verb: http://nihognodaybydayenglish.blogspot.com/2011/03/aku-hiraku.html