Vim doesn't support rainbow parentheses highlighting, out-of-the-box, for anything except Lisp. Rainbow highlighting is pretty awesome though, so if you want it for something like C or C++, a plugin is your best option. I had been using kien's plugin for the last few years, and was generally happy with it. While updating my plugins, though, I noticed that kien's plugin actually hadn't been updated in quite some time, despite having open bugs / pull-requests. After looking around a bit, I found that there are quite a few different rainbow parentheses plugins to choose from. Here is a quick chart to help you choose which one to use.

N.B. You can easily enable lisp rainbow highlighting by putting this in your vimrc:

let g:lisp_rainbow = 1  

Choosing a Rainbow Parentheses Plugin

The following are listed in order of my recommendation.

[Update: When I first posted this list, there were a number of actively maintained options. Since then, the number of active RP plugins has dropped, and ViM users seem to be coalescing around just a few of them.]

Plugin Maintained Last Updated Notes
luochen1990's rainbow Yes Jun 2015 Active maintainer, operator highlighting, very fast
kien's rainbow_parentheses ??? Mar 2013 Does not appear to be maintained, supports chevron hightlighting, generally stable.
oblitum's rainbow No May 2014 Abandoned, has chevron hightlighting, maintainer recommends using luochen1990's plugin
Martin Krischik's rainbow_parenthsis No Oct 2007 Not maintained, supports chevron hightlighting, basis for many of the above plugins.
John Gilmore's RainbowParenthsis No Mar 2005 Not maintained, was the original rainbow hightlighting plugin for Vim
amdt's vim-niji Null Null Repo completely blown away by the maintainer

As you might have guessed from the table above, I'm using louchen1990's plugin. I actually filed a couple of bugs against it when I first started using it, and the maintainer addressed them within a couple of days. The only thing I wish it had is chevron highlighting, which is really great for C++ code.