Elder and eldest mean the same as older and oldest. We only use the adjectives elder and eldest before a noun (as attributive adjectives), and usually when talking about relationships within a family:

Let me introduce Siga. She’s my elder sister.

Not: … She’s my sister. She’s elder.

It’s Catherine’s eightieth birthday on Thursday and she is now the eldest member of the family.

We also use elder and eldest as nouns:

Jack is the eldest of four brothers.

Older and oldest are used in similar ways:

My older sister is coming to stay with us at the weekend.

Matt is the oldest of our children and Simon is the youngest.

Older and oldest can be used to refer to the age of things more generally:

The town hall is by far the oldest building in the whole region.

Not: The town hall is by far the eldest building …

We can use older and oldest after a linking verb (as predicative adjectives):

I think her grandfather must be older than her grandmother.