It's called Screen Guide, and apparently it was printed for at least a decade and a half; the earliest issue I could trace was from June 1936, while the latest was dated November 1951. Beyond that, I don't know much about the publication, such as what company printed it (I have never seen an actual copy of the magazine).
As you might expect, during Lombard's lifetime she occasionally appeared in Screen Guide. In fact, that June '36 issue contains a two-page spread detailing how one of Carole's glamour shoots came about, and the pictures appear pretty fascinating:
For at least part of its run, Screen Guide was "oversized," that is, printed on 10-by-13 inch paper similar to Life and other photo-oriented magazines of the era. Most film fan magazines of the time were printed on smaller stock and focused on stories rather than photos.
Here's an example of what I mean -- the October 1938 issue, featuring Lombard on the cover:
Both of the above items are now available at eBay.
For the June 1936 issue, which has Don Ameche on the cover and an attractive Jean Harlow portrait on the inside back cover, go to http://cgi.ebay.com/Screen-Guide-Jean-Harlow-Carole-Lombard-Mae-West-1936_W0QQitemZ160307454496QQcmdZViewItemQQptZMagazines?hash=item160307454496&_trksid=p3286.c0.m14&_trkparms=66%3A4%7C65%3A1%7C39%3A1%7C240%3A1318. It'll be available through 10:45 p.m. (Eastern) next Tuesday, Jan. 6; no bids have been placed as of this writing, and bidding begins at $5.95.
As for the October 1938 issue, which includes the stories "The Truth About Hollywood Nose Operations" (was everyone trying to emulate Myrna Loy?) and "What Is Sex Appeal Worth To A Movie Actress?", check out http://cgi.ebay.com/Screen-Guide-Movie-Magazine-Oct-1938-Carole-Lombard_W0QQitemZ330297508532QQcmdZViewItemQQptZMagazines?hash=item330297508532&_trksid=p3286.c0.m14&_trkparms=66%3A4%7C65%3A1%7C39%3A1%7C240%3A1318. Bids begin at $9 (none have yet been placed), and the deadline is 5:15 p.m. (ET) next Thursday.
Here's one more example of Lombard and Screen Guide -- the November 1937 cover (which currently isn't up at auction):
If anyone knows more about the history of this magazine, please share it with us.