How to calculate the number of occurrence of a given character in each row of a column of strings?

The stringr package provides the str_count function which seems to do what you're interested in

# Load your example data
q.data<-data.frame(number=1:3, string=c("greatgreat", "magic", "not"), stringsAsFactors = F)
library(stringr)

# Count the number of 'a's in each element of string
q.data$number.of.a <- str_count(q.data$string, "a")
q.data
#  number     string number.of.a
#1      1 greatgreat           2
#2      2      magic           1
#3      3        not           0

If you don't want to leave base R, here's a fairly succinct and expressive possibility:

x <- q.data$string
lengths(regmatches(x, gregexpr("a", x)))
# [1] 2 1 0

nchar(as.character(q.data$string)) -nchar( gsub("a", "", q.data$string))
[1] 2 1 0

Notice that I coerce the factor variable to character, before passing to nchar. The regex functions appear to do that internally.

Here's benchmark results (with a scaled up size of the test to 3000 rows)

 q.data<-q.data[rep(1:NROW(q.data), 1000),]
 str(q.data)
'data.frame':   3000 obs. of  3 variables:
 $ number     : int  1 2 3 1 2 3 1 2 3 1 ...
 $ string     : Factor w/ 3 levels "greatgreat","magic",..: 1 2 3 1 2 3 1 2 3 1 ...
 $ number.of.a: int  2 1 0 2 1 0 2 1 0 2 ...

 benchmark( Dason = { q.data$number.of.a <- str_count(as.character(q.data$string), "a") },
 Tim = {resT <- sapply(as.character(q.data$string), function(x, letter = "a"){
                            sum(unlist(strsplit(x, split = "")) == letter) }) }, 

 DWin = {resW <- nchar(as.character(q.data$string)) -nchar( gsub("a", "", q.data$string))},
 Josh = {x <- sapply(regmatches(q.data$string, gregexpr("g",q.data$string )), length)}, replications=100)
#-----------------------
   test replications elapsed  relative user.self sys.self user.child sys.child
1 Dason          100   4.173  9.959427     2.985    1.204          0         0
3  DWin          100   0.419  1.000000     0.417    0.003          0         0
4  Josh          100  18.635 44.474940    17.883    0.827          0         0
2   Tim          100   3.705  8.842482     3.646    0.072          0         0