The conventional contraction of "semiprofessional magazine".
Starting in the 70s, a semiprozine was a low-paying (low-paying even by the dreadful rates paid by the prozines!) fiction magazine which published mostly amateur fiction too good to be fanfic and too bad to be bought by the prozines. Thus, there was a continuum between fanzines which primarily published fanfic and amateurish attempts at stories (practically never paying), semiprozines which sometimes paid a nominal amount and thus got the cream of amateur sf, and the actual prozines which paid something resembling professional rates. The exact location of the dividing lines between the categories are completely arbitrary, but most fans feel that three categories better express the reality than would two or four.
Such semiprozines were generally lumped in with fanzines in fannish minds and when it came to the Hugo Awards. But in the 70s, non-fiction magazines such as Locus, Science Fiction Chronicle, and Science Fiction Review, all fanzines by the Hugo rules of the time, came to dominate the Best Fanzine Hugo: between 1969 and 1983, it was only once won by a "traditional", fannish fanzine. This was widely perceived as unfair, since while those magazines were like older newszines in content, they had circulations in the thousands, enough to make profit for their publishers. Whatever it was that they were, they were no longer recognizable as fanzines which had always been a labor of love without financial considerations. Equally, they were not recognizably prozines which still dominated the short fiction field (and had no Hugo category per se, it having been replaced with Best Professional Editor Hugo).
The solution adopted (though not without arguments and protests) in 1982 (i. e. with effect from 1984) was to recognize a new category of magazine called Best Semi-Professional Magazine Hugo – which mouthful always becomes Semiprozine – with complex and somewhat flaky boundaries separating semiprozines from fanzines and prozines. Initially, the main criterion was basically the circulation: under 1000 was a fanzine and over 10,000 was a prozine (though it was really much more complicated, see Best Semiprozine Hugo). By 2010, however, the rise of fiction webzines had made circulation meaningless and the dividing line became basically financial. See WSFS rules governing the Hugo Awards for details.