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 creme 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.
Into the 70s, semiprozines were generally lumped in with fanzines in fannish minds and when it came to the Hugo Awards. But in the 70s and 80s, non-fiction magazines such as Locus, SFC, and Science Fiction Review (all fanzines by the Hugo rules of the time) came to dominate the Best Fanzine Hugo category. This was widely perceived as unfair, since while those magazines were like traditional fanzines in content, they had circulations in the thousands and were becoming profitable enough to pay salaries for their staff. Whatever it was that they were, they were no longer recognizable as fanzines which had always been a labor of love and not for financial gain. Equally, they were not recognizably prozines which still dominated the short fiction field.
The solution arrived at was to recognize a new category of magazine called Semi-Professional Magazine -- 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 more complicated than that). By 2010, however, the rise of online fiction outlets had made circulation meaningless and the dividing line became basically financial. See WSFS rules governing the Hugo Awards for details.
See also: Amanita Brandy - Andy Porter - Ansible - Arnie Fenner - Axel Melhardt - Best Editor, Long Form Hugo - Best Editor, Short Form Hugo - Best Semiprozine Hugo - Black Static - Broken Mirrors Press - Clarkesworld - Crank - Dale C. Donaldson - David G. Hartwell - Fantastic Worlds - Fantasy Magazine - Fantasy Review - Fantasy Review - Galaxy - Interzone - Jerry Siegel - John Klima - Kees van Toorn - Kelly Link - Lady Churchill's Rosebud Wristlet - Midnight Graffiti - Miriad - Moonbroth - NCSF - Omnifan - Pandora - Pirate Writings - Prelude to Fantasy - Publications - Quantum - Rigel Science Fiction - Robert E. Howard Companion - Science Fiction and Fantasy Book Review Annual