Scholarships for Indian students are generally of the following types:
a) Funded by the Department
b) Funded by the University
c) Funded by the college ( the system in UK)
d) Funded by government and other bodies in India/ International bodies having national selection in India.
For (a) and (b),often you are automatically considered when you fill the university application form. Certain application forms have special sections which you need to fill only if you want scholarship (eg: Gates Cambridge Scholarship). The deadline for such scholarship application generally co-indices with the deadline for the application form. In certain cases, however, the scholarships applications have earlier deadline meaning thereby that if you want to be considered for the scholarship, you have to submit your application form earlier. Do always check the funding section of university website and department website. There might be scholarships for which you have to fill separate forms. Also, some universities (like London School of Economics and Harvard) have a financial need form.
In UK, certain universities (like Oxford) have scholarships offered by colleges only (c) and you are eligible for such scholarships only if you are part of that college. These scholarship applications are often different from University or Department scholarship applications and have to be filled out separately.
Examples of (d) are Rhodes , Commonwealth, Inlaks, Oxford Cambridge Society of India etc. These scholarship applications generally open in February, Rhodes and Commonwealth being notable exceptions.
In addition, there can be sponsorships by individuals, offices etc.