I can hear everybody saying that “Web Analytics Service” under SharePoint 2013 has gone and has been redesigned under Search Services. Actually Web Analytics is not gone; it has only changed its clothes.  
As many of yours know; the feature "Web Analytics" in SharePoint 2010 has not been renewed on SharePoint 2013. 
For the one who don’t know about what I’m talking: This was the service under SharePoint 2010. I’m a little bit nostalgic today.

Analysis of the data has been completely redesigned and is now a component Search Service Application: it analyzes the content and the actions taken by users on the site of the content.

This information uploaded by the analysis is then injected into the index to improve the relevance of research.

What’s changed ?

• Report « Top items » has been kept but reports here under were deleted from the application service:

•  Web Part « Web Analytics » is not supported in SharePoint 2013 anymore
•  Report « Browser traffic »,
•  Report « Top Users »,
•  Report « Referring URL »,

Migration from SharePoint 2010 
Many people want to migrate from SharePoint 2010 to SharePoint 2013. But I can confirm it is not possible to migrate the  service application "Web Analytics" from SharePoint 2010 to SharePoint 2013.
It is useless to try to recover your databases for SharePoint 2010 to SharePoint 2013 component, it will not use your meter and whatever happens will leave the counter to 0.

Advantages

•  The new analytics engine finds relevant information based on clicks, views, etc.
•  You can get hot indicators and usage numbers based on number of views and number of unique visitors
•  You can understand how much content has been used
•  This engine is extensible for 3rd parties
•  Counting clicks / views for each document
•  Recommendation of content
•  Search results influenced by the priority of an item
•  Ability to sort the results by "hit"
• 
  

The search recommendations framework works in the following way:

• When users interacts with a SharePoint Server 2013 — for example, when the users  clicks a link, presses a button, or views a document — actions are stored as usage events.

• Usage events are counted and analyzed. The recommendations algorithm in the Analytics Processing Component counts and analyzes the usage events.

Information is added to the index. After processing in the Analytics Processing Component, the information is added to the search index and the Reporting database.

Source: spdailytips.blogspot.com

How does it work?

1. Under Central Administration open your Search Service Application.

2. At the left side clique on « Usage Reports ».

3. You can now see a lot of reports that you can extract and see information:

Explanation of each report:

 Number of Queries
This report shows the number of search queries performed. Use this report to identify search query volume trends and to determine times of high and low search activity.

 Top Queries by Day
This report shows the most popular search queries. Use this report to understand what types of information visitors are seeking.

 Top Queries by Month
This report shows the most popular search queries. Use this report to understand what types of information visitors are seeking.

 Abandoned Queries by Day
This report shows popular search queries that received low click-through. Use this report to identify search queries that might create user dissatisfaction and to improve the discoverability of content. Then, consider using query rules to improve the query's results.

 Abandoned Queries by Month
This report shows popular search queries that received low click-through. Use this report to identify search queries that might create user dissatisfaction and to improve the discoverability of content. Then, consider using query rules to improve the query's results.

 No Result Queries by Day
This report shows popular search queries that returned no results. Use this report to identify search queries that might create user dissatisfaction and to improve the discoverability of content. Then, consider using query rules to improve the query's results.

 No Result Queries by Month
This report shows popular search queries that returned no results. Use this report to identify search queries that might create user dissatisfaction and to improve the discoverability of content. Then, consider using query rules to improve the query's results.

 Query Rule Usage by Day
This report shows how often query rules trigger, how many dictionary terms they use, and how often users click their promoted results. Use this report to see how useful your query rules and promoted results are to users.

 Query Rule Usage by Month
This report shows how often query rules trigger, how many dictionary terms they use, and how often users click their promoted results. Use this report to see how useful your query rules and promoted results are to users.

Source: Microsoft.com

This is an example about a report that you can get:

How does it work under a site:
1. Connect to your site and click on « Site Settings ».
2. On site level click « Popularity Trends ».

3. There is only one report available please click on it:

4. on site Collection Level, click on « Popularity and Search Reports ».

5. Ten reports are available now just select one and open your report.

How does it work under a library:

1. Go to your library and click on « Most Popular items »

2. You can now select a report :

Google Analytics

So if you do not want to use the native “Web Analytics” of SharePoint 2013 there is another way to get statistics from your site. 
The solution is based on the Google Analytics solution for SharePoint 2010. It enables on all pages the Google Analytics code or some other JavaScript without modifying the underlying master pages or any other file delivered from Microsoft. The solution runs as a non-code sandbox solution. That should limit deployment difficulties. 
Download: http://www.fiechter.eu/blog/Solutions/Wsp365.GoogleAnalytics.zip   
Codeplex: http://googleanalytics365.codeplex.com/