This problem has 2 known causes and 3 possible fixes.

First of all, the causes:

  1. Some textures are corrupted
    If this is the case, there will be a "log.txt.0" file in the "logs" folder within the install folder, ending with an Exception stating some resolutions don't match.
    It can easily be fixed by overriding the install using the force-override function in the download options within the launcher
  2. The JVM fails to launch
    If this is the case, and the install is fresh, there will be no file in the "logs" folder, if such a folder exists within the install folder.
    This is caused by one of the 2 known issues:
    1. 32 bit java is used on a 64 bit OS
      This can easily be fixed by installing 64 bit java instead. If 64 bit java is already installed while 32 bit java is present, make sure 64 bit java is set as default.
    2. The JRE (Java Runtime Environment) does not provide a required function on your setup
      This can be fixed by installing the JDK (Java Development Kit) instead, since it is supposed to provide every function in every environment. You can get it here.

If none of these fixes can resolve your problem, please leave a ticket.