I hold both a Canadian & British Passport. Eligible for the British through my Father (brithish by decent).
When I applied for my British passport, I needed:
1) a copy of my dad's birth certificate, proving he was born in the UK (it didn't matter, that he holds a Canadian Citizenship now & has actually been in Canada longer than he was in the UK). I had to get in touch with some records office in the UK as he didn't have his birth certificate, took a couple of weeks.
2) a copy of my full birth certificate, one that indicated both my parents names. the birth certificate I have is just the 'short' one, so I had to contact the Ontario government to get this
3) a copy of my parents wedding certificate, showing that they were married at the time of my birth. If they weren't married, I'd only be eligible for right of abode & not citizenship / passport. As my parents had divorved several years ago, I had to get this out of the UK too.
4) the actual passport application was pretty straightforward, once all the paperwork was gathered.
hope this helps.