Recursively count all the files in a directory

find . -type f | wc -l

Explanation:
find . -type f finds all files ( -type f ) in this ( . ) directory and in all sub directories, the filenames are then printed to standard out one per line.

This is then piped | into wc (word count) the -l option tells wc to only count lines of its input.

Together they count all your files.

For files:

find -type f | wc -l

For directories:

find -mindepth 1 -type d | wc -l

They both work in the current working directory.

With bash 4+

shopt -s globstar
for file in **/*
do
  if [ -d "$file" ];then
    ((d++))
  elif [ -f "$file" ];then
     ((f++))
  fi
done
echo "number of files: $f"
echo "number of dirs: $d"

No need to call find twice if you want to search for files and directories

Slight update to accepted answer, if you want a count of dirs and such

find $DIR -exec stat -c '%F' {} \; | sort | uniq -c | sort -rn