There are 3 built-in file descriptors: stdin, stdout and stderr (std=standard). (You can define your own, see the linked article.)

Basic

• 0/1/2 references stdin/stdout/stderr
• >/2> redirects stdout/stderr, where > is taken as 1>
• &1/&2 references stdout/stderr
• &> redirects stdout and stderr = everything (caution: see below)

Caution: &> is functional as of Bash 4 Show archive.org snapshot . This seems to result in a slightly differing behaviour when redirecting output in Ruby scripts.
Instead of cmd &> file, prefer cmd > file 2>&1, which equals: "Redirect stdout of cmd to file, and redirect stderr just where stdout is going", e.g. command > /dev/null 2>&1.

Applied to files

• > creates or overwrites a file
• >> creates or appends to a file
• < reads from a file, where < is taken as 0<

Note: You should be pretty sure of what a command is doing if you are going to wipe it's output.

Redirection Examples

• stdout to file: ls -l > ls-l.txt
• stderr to file: grep da * 2> grep-errors.txt
• one to another: grep da * 1>&2
• everything: do_stuff &> /dev/null
• combined: command < input-file > output-file, reads from input-file and writes stdout to output-file