New Relic is a monitoring tool that lets you see how WordPress is performing, and can warn you if your web site is running slowly, for example. You don't need it to run W3 Total Cache, but it can provide useful data.
You'll need to set up an account at newrelic.com - a free account should suffice. Once you have an account, you will also need to install some software on your server, which is New Relic's PHP Agent. Instructions for this are at https://docs.newrelic.com/docs/agents/php-agent . You will also need an API key from New Relic which you can find in your account settings. Paste the API key into the W3 Total Cache settings and you should be good to go.
There's a step-by-step guide that explains this in more detail at http://code.tutsplus.com/tutorials/using-new-relic-to-monitor-wordpress-performance--cms-22002 . Good luck!