Predatory publishers and journals promise academic publishing opportunities for a fee without actually performing those academic publishing functions. Their goal is largely to make money off of publication fees that they gather from authors.
It is important to note that predatory publication happens on a spectrum. On one end are publishers/journals that trick authors into paying fees without ever "publishing" their work or even holding research data hostage until they receive a certain amount of money. On the other end are publishers/journals who may provide some publication functions, like actually putting author work online, but have lax or low quality scholarship standards.
Note on fees: A publication fee does not equal predatory behavior. Many reputable publications require a publication fee if you want your work to be open access. Publishers that have open access options forgo money that they would usually receive through more traditional subscription based fees. Because of this, publishers do need to require fees to cover costs of providing the work for "free." Predatory publishers take advantage of this fee-based system. This is why it is important to not use the presence of fees as the determining factor of predatory behavior.
The following are quick checks that you can do that can indicate whether you need to do more digging to determine whether a publisher is predatory.
These are not perfect checks and should be only used as a first step to point you in the right direction.
If the journal you are looking up is a health sciences journal, a good place to look is PubMed.
You can also check Boxer Library journals.
Check if the journal or publisher is listed on Beall's List. Beall's List was started by Jeffery Beall in an attempt to list publishers/journals that had suspicious behavior. This list is no longer active and is considered archived. Like many other tools in this process, this is not a perfect list. Some publishers or journals have refuted the fact that they are on the list. Inclusion on the list should only serve as an indication that you should investigate the publisher further.
Simply type the publisher or journal name into Google and see if there are any posts or questions from other scholars regarding its behavior. Sometimes you will find things on forums like Reddit, Quora, and StackExchange. These forums may help you see if other scholars are being solicited with the same emails or tactics.
Published journals should have an ISSN (International Standard Serial Number) assigned to them. You can check this in the International Standard Serial Number Portal. ISSN's are an eight digit number divided by a hyphen into two four digit parts. Example: 1234-1234
Be aware that some journals will make up an ISSN to put on their website (which you can double check through the ISSN portal).