Next day vs the next day

'Next day' rather means tomorrow, whereas 'the next day' is used to talk about the day after a particular day you are talking about, especially in the past context.
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He came from America on June 12, 2008. The next day, he went missing.
I will come next day. (tomorrow)

In both cases the preposition of time (on) is usually omitted.

Next-day

Next-day is used as an adjective
Next-day delivery
Next-day appointment