R group usually means an alkyl group.
Examples of alkyl groups are methyl (-CH3), ethyl (-CH2CH3), propyl (-CH2CH2CH3), butyl (-CH2CH2CH2CH3), etc.
Of course, apart from straight-chained alkyl groups, you can have branched alkyl groups; for instance, 2-methyl butyl (-C(CH3)) is a branched R group (imagine the middle carbon bonded to 3 methyl groups).
Why do we use R-groups?
R groups are a way of representing a class of organic compounds. For instance, alcohols are represented as R-OH, where OH is a functional group.
Thus, if we let R to be methyl or -CH3, R-OH simply represents methanol.
The study of how compounds interact with one another is represented using mechanisms. When the R group becomes large (eg 20 carbons), the use of R groups simplifies mechanisms by leaving out non-reactive part of the compounds.
Can R-groups involve non-carbon or non-hydrogen atoms?
Yes by common practice, though technically it shouldn't.