How do I use bash's "<<" operator?

I know bash has a >> operator, to append output to files, but what is the use of the << operator?

Use

info bash

and then search (type '/' followed by '<<') .
Here's a partial quote:

Here Documents
   This type of redirection instructs the shell to  read  input  from  the
   current source until a line containing only delimiter (with no trailing
   blanks) is seen.  All of the lines read up to that point are then  used
   as the standard input for a command.

   The format of here-documents is:

          <<[-]word
                  here-document
          delimiter

   No  parameter expansion, command substitution, arithmetic expansion, or
   pathname expansion is performed on word.  If any characters in word are
   quoted,  the  delimiter is the result of quote removal on word, and the
   lines in the here-document are not expanded.  If word is unquoted,  all
   lines  of  the here-document are subjected to parameter expansion, com‐
   mand substitution, and arithmetic expansion.  In the latter  case,  the
   character  sequence  \<newline> is ignored, and \ must be used to quote
   the characters \, $, and `.

   If the redirection operator is <<-, then all leading tab characters are
   stripped  from  input  lines  and  the line containing delimiter.  This
   allows here-documents within shell scripts to be indented in a  natural
   fashion.

Here Strings
   A variant of here documents, the format is:

          <<<word
   The word is expanded and supplied to the command on its standard input. [...]

$ cat <<EOF
> This is a test
> EOF
This is a test
$