These three terms essentially mean the same thing. Bursary is the term for scholarship used in the UK. Normally, a bursary is given based on need which is not always the case with fellowships/scholarships.  In my personal experience in America, the term bursary is rarely used. Instead, we often use the term "grant", which is a synonym for bursary.  

Fellowship is also another term for bursary/scholarship. However, fellowships are usually awarded at the graduate level whereas a scholarship can be awarded at either grad or undergrad level. In addition, fellowships are often considered a highly prestigious award.

As you can see, there are minor differences in these three terms. Determining which to use will mainly depend on the context of the situation