To allow folds based on syntax add something like the following to your .vimrc:

set foldmethod=syntax
set foldlevelstart=1


let javaScript_fold=1         " JavaScript
let perl_fold=1               " Perl
let php_folding=1             " PHP
let r_syntax_folding=1        " R
let ruby_fold=1               " Ruby
let sh_fold_enabled=1         " sh
let vimsyn_folding='af'       " Vim script
let xml_syntax_folding=1      " XML

Syntax based folding is defined in the syntax files of the language which are located in $VIM/syntax or /usr/share/vim/vimXX/syntax/. But some languages do not have folding rules built into their syntax files; for example Python. For those languages you need to download something from http://vim.sf.net that does folds. Otherwise you will need to use folds based on indents. To do this effectively you will likely want to add the following to your .vimrc file:

set foldmethod=indent
set foldnestmax=2

Other kinds of folding

There are 6 types of folds:

manual          manually define folds
indent          more indent means a higher fold level
expr            specify an expression to define folds
syntax          folds defined by syntax highlighting
diff            folds for unchanged text
marker          folds defined by markers in the text

Personally, I only use syntax folds. Usually, I just want to fold the method and not fold every indent level. Inconsistent indenting and weirdly formatted legacy code at work often makes indent folding difficult or impossible. Adding marks to the document is tedious and people who do not use Vim will not maintain them when they edit the document. Manual folds work great until someone edits your code in source control and all your folds are now in the wrong place.

More reading

1. See :help fold-methods to learn the details of different fold methods.
2. See :help folding to learn the keyboard commands for manipulate folds.
3. See :help folds for help on the entire topic of folding.

JavaScript folding didn't work for me either. I found out when I did set syntax=javaScript (with a capital S), it suddenly worked.

The way to enable folding in new versions of Vim has changed (I'm using vim 7.4). Now you should create the file ~/.vim/ftplugin/javascript.vim (on linux) and add your code folding instructions as explained in Eric Johnson's answer. File type detection and loading plugins for specific file types must be enabled by putting this into your .vimrc:

filetype plugin on

Tried all of the solutions here and none worked with NeoVim v0.3.1 Until I found the vim-javascript plugin and folding started to work.