someday: describes an indefinite future time ('I'd like to see him again someday')
some day: refers to a single day ('I have an appointment some day next month')
the other day: means a few days ago ("The other day I saw her downtown")
another day: an action is not going to be finished today ('I'll finish the book another day')
the other day: means a few days ago ("The other day I saw her downtown")but one day he wans't there'/ 'One day this world will be a perfect place to live') * "one day" is more dramatic than "someday"
all day long: the whole day ('We waited for you at the station all day long')
all in a day's work: of a task, normal and ordinary ('Dealing with screaming kids is all in a day's work when you are a pediatrician')
any day: any time, no particular time ('I don't need the report back immediately, any day will do')
any day now: some day very soon, imminently ('We're expecting a letter from her any day now')
at the end of the day: ultimately ('At the end of the day, it's your decision and nobody else's')
by the day: one day at a time, per day ('I rent this room by the day')
by day: during the day ('By day, Mary worked in an office, by night, she took classes')
call it a day: to stop working ('I'm tired, let's call it a day')
day by day: daily, as the days pass ('Day by day she grew more confidente about the job')
day in and day out: everyday ('They eat nothing but vegetables, day in, day out')
day-to-day: daily, regularly occurring, current ('And you'll oversee the company's day-to-day operations while I'm travelling')