%>% has no builtin meaning but the user (or a package) is free to define operators of the form
%whatever% in any way they like. For example, this function will return a string consisting of its left argument followed by a comma and then it's right argument.
"%,%" <- function(x, y) paste(x, y, sep = ",") # test run "Hello" %,% "World" ##  "Hello,World"
The base of R provides
%*% (matrix mulitiplication),
%/% (integer division),
%in% (is lhs a component of the rhs?),
%o% (outer product) and
%x% (kronecker product). It is not clear whether
%% falls in this category or not but it represents modulo.
magrittr In the case of
%>% the magrittr R package has defined it as discussed in the magrittr vignette. See http://cran.r-project.org/web/packages/magrittr/vignettes/magrittr.html David Arenburg also provided this link in the comments: http://rpackages.ianhowson.com/cran/magrittr/man/pipe.html
magittr has also defined a number of other such operators too. See: http://cran.r-project.org/web/packages/magrittr/magrittr.pdf
dplyr The dplyr R package used to define a
%.% operator which is similar; however, it has been deprecated and dplyr now recommends that users use
%>% which dplyr imports from magrittr and makes available to the dplyr user. As David Arenburg has mentioned in the comments this SO question discusses the differences between it and magrittr's
%>% : Differences between %.% (dplyr) and %>% (magrittr)
pipeR The R package, pipeR, defines a
%>>% operator that is similar to magrittr's %>% . See http://renkun.me/pipeR-tutorial/
The pipeR package also has defined a number of other such operators too. See: http://cran.r-project.org/web/packages/pipeR/pipeR.pdf
expm The R package, expm, defines a matrix power operator
%^%. For an example see Matrix power in R .
operators The operators R package has defined a large number of such operators such as
%!in% (for not
%in%). See http://cran.r-project.org/web/packages/operators/operators.pdf
Update Added info on expm package and simplified example at top.