There are various descriptions of the terms, and unfortunately blogs, tutorials, enterprise framework documentation and diagramming software tend to conflate the concepts. For more rigorous definitions, consult academic papers and books by the founders of the field.
An entity is a thing which can be distinctly identified, like a specific person, company, or event. Entities are identified by values in a database, e.g. I (an entity in the real world) am represented by the number 532721 in StackOverflow's database.
An entity set is a set of similar things, like a set of persons, companies or events. An example would be all the users on StackOverflow. Entities and entity sets are conceptual and not directly contained in databases. StackOverflow's database talks about its users, those users don't actually live in the database.
A table is a data structure which represents a predicate. A predicate is a fact type, a generic statement with placeholders for values. Records contain values for those placeholders that make the predicate true, thus records represent propositions about entities in the world. Another way to view it is that a table represents a set of attributes and relations on one or more entity sets. Remember attributes are just binary relations.
For example, a table USER (UserId PK, UserName UQ, Reputation, PhotoId UQ) can be understood as saying "There exists in the world a user identified by a number UserId and unique name UserName who has a score of Reputation points and exclusively uses the photo identified as PhotoId as avatar". Each corresponding record represents a known fact about a user and an image.
I recommend you read Codd's paper "A Relational Model of Data for Large Shared Data Banks" and Chen's paper "The Entity-Relationship Model - Toward a Unified View of Data". They're shorter and more focused than a whole book, and can be easily found online.
