Slots are linked to S4 objects. A slot can be seen as a part, element or a "property" of an object. Say you have a car object, then you can have the slots "price", "number of doors", "type of engine", "mileage".
Internally, that is represented a list. An example :
Here, price, numberDoors, typeEngine and mileage are slots of the S4 class "Car". This is a trivial example, in reality slots themselves can be again complex objects.
or through the construction of a specific method (see extra documentation).
For more on S4 programming see this question. If the concept still sounds vague to you, a general introduction in Object Oriented Programming could help.
PS: Mind the difference with dataframes and lists, where you use $ to access named variables/elements.
In addition to the resources @Joris points you to, plus his own answer, try reading ?Classes, which includes the following on slots:
Slots:
The data contained in an object from an S4 class is defined
by the _slots_ in the class definition.
Each slot in an object is a component of the object; like
components (that is, elements) of a list, these may be
extracted and set, using the function ‘slot()’ or more often
the operator ‘"@"’. However, they differ from list
components in important ways. First, slots can only be
referred to by name, not by position, and there is no partial
matching of names as with list elements.
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