Counting files (list and word count, ls & wc)

How do you count the number of files in a directory? That becomes non-trivial if you have thousands of files in a directory. Your file manager chokes on counting them, especially if they are not local. On the Linux command line, this task is fast and easy:
ls | wc
ls lists your files one per row and wc returns three numbers: lines, words and bytes. In the file counting example, lines and words are the same and equal the number of files in the directory.