Most of the following are aimed at undergraduate and masters students, those who want to see what real-life research is like, and dip their toes into a field they think that they might like to pursue.
For Europe and the UK click here.
For Canada and the US click here.
AAO – The Australian Astronomical Observatory offers Student Fellowships both in the Australian Winter and Summer months, for 10-12 weeks, with A$700 per week.
Swinburne – Vacation scholarships at the Centre for Astrophysics and Computing, at Swinburne University of Technology. Mostly for Australians and Kiwis, however exceptional international students may apply. Usually over summer (i.e. winter for most of us), 6 – 10 weeks, A$500 per week.
TAU – Scholarships are available to go to Israel and carry out research for 8 months, for those with a BSc. For research ideas/contacts take a look at The Raymond & Beverly Sackler School of Physics & Astronomy at the Tel Aviv University. There is also the Nella and and Leon Benoziyo Center for Astrophysics at the Weizmann Institute of Science, and not forgetting Technion, the Israel Institute of Technology.
JAXA – Internships available at the Japan Aerospace Exploration Agency in Japan.
JSPS – The Japan Society for the Promotion of Science has a two month summer programme.
UTRIP – An intensive summer research programme at the University of Tokyo.
ASIAA – A summer student programme in Taiwan.
OAD – Based in Cape Town, one has the chance to get a Fellowship for the Office of Astronomy for Development. Flights, accommodation and daily subsistence covered!
SGAC – The Space Generation Advisory Council often has internships and volunteering roles available.
SSP – The Space Studies Program is a two month summer programme, with disciplines including space physical sciences; space engineering; space policy, economics and law; space management and business; space and the humanities; space applications; and human performance in space. In 2016 it will be in Haifa, Israel, and in 2017 it will be in Cork, Ireland.
SH-SSP – The Southern Hemisphere Space Studies Program lasts for 5 weeks, and will be in Australia in 2016.
Don’t however limit yourself to specific summer schools/internships if you feel like you are looking for something different. For example, you could volunteer. On the other hand you might want to work for someone at a particular university, and you may be able to find funding from a trust for example.