A scholarly source is publication written by an expert in a particular field and published in an academic journal. A peer-reviewed source is a particular kind of scholarly source that goes through a rigorous review process before publication.
An article in a magazine is not scholarly. For example, an article in Time magazine is written by a journalist, often one employed by the magazine, who may be assigned to write on a variety of topics. The journalist's area of expertise is in writing, not in the fields about which he or she writes.
Academic journals do not employ journalists. The articles they publish are submitted by experts. Economists might submit articles to an academic journal of economics, a biologist to an academic journal of biology, and so on.
A peer-reviewed journal is a special kind of academic journal that submits every article submitted to a rigorous review process before publishing it. Taking our above example of a biology journal, a biologist might submit an article about an experiment to the editors of a peer-reviewed biology journal. The editors will then send that article out to other expert biologists who check it to make sure the experiment is reproducible and the conclusions are valid. If and only if the article passes the peer-review, then it will be published.
Most of the library's databases have a check box to make it easy to search for only "peer-reviewed" articles. It will look something like this:
