Learn Turkish: Terziye yeni bir elbise diktirdim. - I had a new dress sewn by the tailor.

It comes from the verb root dik- (to sew). We add the causative suffix -tir to make diktir- (to have [something] sewn). Then we add the past tense -di and the first-person ending -m. The causative suffix is normally -dir, but it changes to -tir here because dik- ends in a voiceless consonant (k).

Why does terzi (tailor) take the dative case ending in terziye?

In Turkish causative sentences, the person you are having do the action takes the dative case (-(y)e or -(y)a). You are the subject, the dress is the object, and the tailor is the agent doing the actual work, so it becomes terziye (literally "to the tailor").

Why doesn't elbise (dress) take the accusative case ending (elbiseyi)?

Because it is an indefinite object. You are having "a new dress" sewn (yeni bir elbise), not a specific, previously mentioned dress. If you wanted to say "I had the new dress sewn," you would use the accusative case and say yeni elbiseyi diktirdim.

I remember dikmek meaning "to plant". Does it also mean "to sew"?