One of the main criteria scholars use for determining the importance of a journal is the number of times articles in a journal get cited (used as references) by authors of other articles. The number of times an article itself gets cited is also a clue to its importance to scholars.
Citation counts become the scoring system for the importance of a journal or an article, particularly in Sciences and Social Sciences-- much less so in Art and Humanities fields of study. That is not to say that important research does not appear in other journals. It is just that research is more likely to be noticed and influence the field when it appears in a journal with a history of publishing highly cited articles because that is where other scholars look first.
The two databases Web of Science and Journal Citation Reports (JCR) are the main tools for finding out what these highly cited journals are. Try Google Scholar, also. It connects you to articles and books citing an article or document along the same lines, but from a much wider, less-selective pool of sources.
Journal Citation Reports (JCR) are the annual scorecards for the journals in Web of Science and can show you the journal titles and rankings of titles in a subject. The Eigenfactor score listed as one of the measures in the report for a journal tries to provide a rating of the total importance of a journal and articles in the journal to a particular field.
Web of Science is limited to the most cited journals. If you find journals in the Journal Lists of Searchable Databases, you can be fairly confident these are prominent journals in their fields.
Web of Science is a good starting point if you want to limit your search on a topic to articles from journals with recognized importance. Web of Science also has search tools from the record of an article, such as Related Records, Times Cited and Citation Map, that help you discover the invisible college of researchers interested in a topic.
Also from this publisher are Current Contents publications, which are related to Web of Science. These list journals that are the most selective and active. See, for example, the journals that appear in Current Contents® / Arts & Humanities Subject Categories and Subject Categories.