Internal transfers are when a student transfers from one of Columbia's schools to another school also at Columbia.
Between CC and SEAS
Students rarely transfer from SEAS to CC, but it's nevertheless possible. To apply for this transfer, a candidate must fill out the same application as transfer candidates from other schools, but the application fee is waived. Supposedly, there is a special internal transfer pool, so internal candidates are not lumped into the application pool from other schools (of which ~5% is accepted). No data is available on how many students successfully attempt to transfer between SEAS and CC per year, but supposedly there were 11 successful candidates in 2003-2004.
As of April 9, 2010, there have been reports that at least 4 students (1 SEAS, 3 CC) from the class of 2013 have successfully transferred.
Also, various sources claim that transfer admission decisions for the 2010-2011 year are to be released on April 23. However, successful internal transfer students have been informed on April 5 and are required to reply to the Office of Undergraduate Admissions by April 12, 2010.
Between CC/SEAS and GS
If you drop out and aren't allowed back as a CC/SEAS student, you may be able to return via GS. There is apparently a procedure for transferring from GS to CC/SEAS but there are no documented cases of this occurring.
Between CC/SEAS and Barnard
At least one student in the class of 2009 transferred from CC to Barnard for the '07-'08 school year.