The main internship advertisement listing is in the American Statistical Association's (ASA) AMSTAT December or January newsletter.  These internship listings (including additional internships added through the spring) are also available at http://www.amstat.org/education/internships.cfm.  Another source of internship information is faculty members.  They sometimes know of "unadvertised" internships.  Most internships have February 1 or later application deadlines (a few are earlier).  A good time to get application materials ready is during the preceding Winter Break.  

Interviews for internships typically involve 1 or 2 phone interviews by a statistician.  Some interviewers only ask general statistical questions like "Have you covered nearest neighbor discriminant analysis in a course before?"  Other interviewers may pry more into your knowledge of statistics with questions like "Could you describe nearest neighbor discriminant analysis and discuss how to implement it?"  

Most internships are only for the summer.  Since internships are often not nearby, companies usually provide some assistance in finding housing.  The compensation for an internship is usually at the level of your corresponding education level.  For example, if you have a MS degree, you generally will get paid at this level.  

While at an internship, take advantage of ALL possible learning opportunities!  Do not spend all of the time in a cubicle isolated from the rest of the company.  Talk to the other statisticians at the company and find out what they do.  Learn about the company and what it does.  Remember that you may be doing a job like this soon so try to determine if this is something that you want to do for the rest of your life!

For more specific information about internships, talk to your major advisor or another faculty member.