In the UK at least, there isn't really a culture of 'internships' in research. You may instead come across the following:
- funded short projects e.g. Wellcome Trust (used to) fund summer projects for the gap between second and third year undergraduate.
- work experience/work shadowing - you would usually have to initiate these yourself by making contact with an appropriate and willing researcher.
- short fixed term contract research jobs.
I cannot comment on the situation in other countries.