1) Objects in R can take various classes like numeric, factor, and logical and know their class through a class attribute.
2) Common data objects in R include vectors (containing a single data type), lists (containing multiple data types), matrices (2D arrays), and data frames (lists of equal-length vectors displayed as a table).
3) The class of an object determines how it can be manipulated and analyzed in R; for example, summarizing numerical versus categorical data requires different functions.