Why there is no vaccine against Human Immunodeficiency Virus (HIV)?
One of the reasons we don't have an HIV vaccine yet is we've only been trying to make one for a little over 30 years. It can take a long time to develop a vaccine. Beyond that, there are a number of specific challenges. 

One of them is that, yes as your teacher said, the virus is highly variable. Specifically, the parts of the viral proteins involved in binding and infecting a host cell are highly variable. There are a number of other challenges. One important and related challenge is that, unlike with many vaccine preventable diseases (e.g., measles), an successful HIV vaccine will have to entirely prevent infection  rather than just control and clear infection without developing disease. Once a reservoir of infected cells is established, the opportunity for clearance is gone. This makes the production and maintenance of high titers of neutralizing antibodies that much more important.

There is a very good review of the challenges and current directions of HIV vaccine development here