The following steps will explain the upgrade process of SharePoint web application from SP 2010 to SP 2013 Farm

Backup the Content database from SP2010 server using SQL management studio

Move the database to SharePoint 2013 SQL server.

Restore the content database on SharePoint 2013 SQL server using SQL management studio

Upgrade custom code from visual studio 2010 to VS 2012 and Version 15 of SharePoint (Recommended). Although, you can still deploy the SharePoint 2010 code on SharePoint 2013 Farm under 14 hive)

Create new web application and root site collection in SharePoint 2013 Farm using “Central Administration”

Deploy custom solutions to the SharePoint 2013 Farm (make sure they are deployed on new web application)

Remove the default content database from the newly created web application using “Central Admin”

Attach the SharePoint 2010 web application content database to new web application in SP 2013 using PowerShell script

Run Test-SPContentDatabase cmdlet to identify missing components along with potential errors and related warnings. Check the upgrade log and deploy any missing components and re run the cmdlet to verify.

Test-SPContentDatabase -name WSS_Content_DB -webapplication http://sitename

Attach the content database to the SharePoint 2013 web application using Mount-SPContentDatabase cmdlet.

//
Mount-SPContentDatabase "MyDatabase" -DatabaseServer "MyServer" -WebApplication http://sitename

Navigate to the site and run visual upgrade (To upgrade the site simply click on “Start now” link on the toolbar, you can also go to Site Upgrade page from Site Setting page as well).

Once the Upgrade complete, your site collection should now be accessible in SharePoint 15 mode

Note: The default authentication type in SharePoint 2013 is “Claim based”. You can still create web application with” Classic authentication” in SharePoint 2013 through PowerShell, but it is recommend using Claim based Authentication in SharePoint 2013 web apps for variety of reasons.  Also, windows Classic mode authentication is deprecated in SharePoint 2013. If your SharePoint 2010 web application is set to classic mode authentication type, then change it to claim based authentication either before or after upgrade web App to SharePoint 2013 Farm.

For more info visit: http://technet.microsoft.com/en-us/library/gg251985.aspx