Web Analytics

2016

Get started with Visual Studio Application Insights
https://azure.microsoft.com/en-us/documentation/articles/app-insights-get-started/

2013

Out of box functionality
official - http://social.technet.microsoft.com/wiki/contents/articles/15468.sharepoint-2013-can-i-choose-between-web-analytics-and-google-analytics.aspx
more
http://en.share-gate.com/blog/new-sharepoint-2013-analytics

https://audministrator.wordpress.com/2014/08/31/get-extensive-web-analytics-reports-from-sharepoint-using-ssrs/

Renamed – Analytics Processing
http://technet.microsoft.com/en-us/library/jj219554.aspx

Web Analytics in SharePoint Server 2010 has been discontinued and is not available in SharePoint 2013. Analytics processing for SharePoint 2013 is now a component of the Search service
http://social.technet.microsoft.com/wiki/contents/articles/17705.sharepoint-2013-web-analytics.aspx

SharePoint Community is not happy about the big step backwards
http://www.collabshow.com/2013/05/23/sharepoint-2013-analytics-a-big-step-backward/

2010

On the detailed web analytics question, it is still on the to-do list, but here are my links

https://social.technet.microsoft.com/wiki/contents/articles/5117.sharepoint-2010-get-web-analytics-summary-for-all-site-collections.aspx

Out of the Box - http://blogs.msdn.com/b/ecm/archive/2010/05/03/web-analytics-in-sharepoint-2010-insights-into-reports-and-metrics.aspx

Logging Database - http://technet.microsoft.com/en-us/library/hh769360(v=office.14)

Good article - http://todd-carter.com/post/2011/04/26/extending-the-sharepoint-2010-health-usage-part-1-feature-and-capability-overview/

Web part – but untested - http://www.nullskull.com/a/1398/sharepoint-list-usage-and-statistics.aspx

3^rd Party - http://www.harepoint.com/Products/HarePointAnalyticsForSharepoint/ReportsList.aspx