As comparative and superlative adjectives, elder and eldest are variants of older and oldest. They also work as nouns referring to, respectively, one who is older or one who is the oldest. The words are usually used in reference to people, especially sons, daughters, sisters, and brothers, and they also tend to connote seniority. They might sound wrong in reference to nonhuman things; for example, we wouldn’t call a car or a house elder than another.
If you’re not sure where to use the words, just keep in mind that except in a few rare uses (church elders, for one), they always bear replacement with older and oldest.
Note that in each of these examples, elder and eldest refer to or describe people:
They all attended Omagh Baptist Church where her father was once an elder. [Telegraph]
I am glad that the first lady and my five children are here this evening on my eldest daughter’s 30th birthday. [Los Angeles Times]
We go to a small village to meet a group of elders. [BBC News]
As the eldest of four children, he suddenly became the family cook. [Toronto Star]
He’s not Bob Dole, the GOP’s elder statesman, making his third bid for the presidency in 1996. [The Atlantic]