Do you mean 'someTHING' or 'someONE'? Because the meaning is different in each case. "Turn someDAY on/down" doesn't make sense.
EDIT: Since you mean 'someONE':
"turn me on" means to excite or arouse. Originally it was more related to drug use, so old songs will use it to mean 'excite' or 'inspire', or as a reference to drugs. But now it is almost always about sexual arousal.
"That dress you are wearing turns me on!"
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"turn me down" means to reject:
"I asked her on a date but she turned me down."