New Relic is a monitoring and alerting tool to help you monitor your applications and servers.
New Relic can help you monitor your application, database or server's performance, uptime and give you insights into them. You can also setup various alerts to ensure uptime.
Yes, New Relic does have a free forever plan with limited functionality.
New Relic setup typically requires setting of their agent software or some libraries to setup data collection. These agents then observe and send data to New Relic servers.
New Relic APM is included in its free for ever plan with one day's historical data.