What makes a transcript official vs. unofficial?
An official transcript is usually sent under some sort of seal, with measures to ensure that the document has not been altered. When delivered to a student, it will usually be in a sealed envelope in such a manner as it will show proof of tampering or opening.

The usual method, though, is for the issuing school to send it directly to the intended recipient. The transcript is then authorized as "official" through some combination of stamping or seal. (Nowadays much of this can be and is handled electronically.)

There may be some schools that have specific rules that would prevent them from accepting what another school sends, but you'd have to look at individual schools' policies to know for sure.