PhD, Post-doc and Summer Fellowships
Forgive my laziness to put the weblinks but google should lead you to them easily. Apart from these, another useful source of information can be the local embassies of the particular country you wish to work in.
For summer internships in India for undergrad and post-grad students, see this Quora page and related links.
Marie Curie fellowships - Post-doc & PhD ; Subjects - All sciences.
Newton Fellowships - Post-doc in UK across all discplines.
EPSRC Fellowships - Post-doc & PhD in UK ; Engineering, Physical and Mathematical Sciences.
Royal Society - Post-doc & PhD in UK across many disciplines.
Royal Commission of Engineering 1851 - Post-doc in Engineering subjects in UK.
Humboldt Fellowships - Post-doc in Germany.
Society in Science – The Branco Weiss Fellowship - Post-doc in Sciences, anywhere in the world.
Erasmus - PhD, Post-doc and Masters programs in Europe.
Paris Graduate School Fellowships.
Berlin Mathematical School
SOME USEFUL WEBSITES :
http://www.dpmms.cam.ac.uk/vacancy/postdoc_opportunities.html - Post-doctoral opportunities in Cambridge & UK.
ON SOCIAL ASPECTS :
Guide for Indian Students Coming to Toulouse (France) (A lot of information holds good for anywhere else in France too)
Technion Indians (A useful page for Indians wanting to visit Technion or Haifa in Israel).