Attaining citizenship based on family history?

Each country has its own rules, you pretty much have to research it for each one. However the easiest way is to google for example "Irish Nationality Law" or "Greek Nationality Law" and look for the wikipedia entry. Usually these do a good job of summarizing the laws. Then you have to research the details and process at the official governmental websites.

As far as I know, most nations won't go past your parent or grandparent to consider eligibility for citizenship. The most liberal policy I am aware of is Ireland, where you can pass citizenship indefinitely as long as you keep registering with the embassy/consulate when the next generation is born. But because many people have not done this the feasible time is to the grandparent (I am an Irish citizen this way; my grandmother was born in Ireland, my mother was therefore an Irish citizen though born in the USA, therefore I became an Irish Citizen as soon as I was registered and so my children will be citizens when they register.

Greece also has a fairly generous law.

Some countries such as Germany or France do not allow it to any real extaent.

It's called citizenship by descent.