Mathematical Reviews is now better known as MathSciNet. It's a subcription-based online database of mathematics research papers and books, produced by the American Mathematical Society. Before the online era, it existed as a paper periodical called Mathematical Reviews that was sent out to libraries and subscribers, but it was never really a "journal"; it only included metadata and not original articles.
An entry in the MathSciNet database contains the bibliographic information for the book or article. For selected items, there may also be a review: a summary of the article written by an independent expert who was recruited by the MathSciNet editors. Despite the name, the "review" doesn't usually evaluate the paper as "good" or "bad" but instead summarizes the contents in the reviewer's own words, sometimes including more background information, context, sources, etc. If there is no review (e.g. the paper wasn't selected for a review, or a reviewer couldn't be found, or the reviewer decided that they had nothing to say) then the database entry includes the article's original abstract instead.
When a journal is indexed by MathSciNet, it means that articles published in that journal are entered into the database. Usually this is cover-to-cover, i.e. including all articles in the journal, but in some cases it could be more selective. MathSciNet has pretty broad coverage and will generally index any journal that seems to be of interest to the community, so being indexed by MathSciNet is a pretty low bar. If a journal is not indexed, it's likely either very new, very obscure, or of such low quality that the MathSciNet editors don't think it is of any significant interest at all.
I don't think it's accurate to speak of a journal being reviewed, since as I understand it, the decision to seek a review is per-article. Even in top journals, not all articles receive reviews (though it's not necessarily clear whether the editors tried to solicit one). Still, presumably papers in strong journals are more likely to be reviewed, as this is some indication that the paper is interesting. In obscure or lower-quality journals, it could happen that none of the articles are reviewed.
So in short, if the publishers say their journal is "reviewed" by Mathematical Reviews, they probably just mean that it is indexed, and that at least some of the papers receive reviews. I would take this as a signal that the journal is probably not complete garbage, but I wouldn't read into it much more than that. As I said, it is a very low bar.