Use "older" to describe the differences in the ages of people (or objects).
E.g., John is older than Mary. The paintings by Rembrandt are older than those by Monet.
My sister has started seeing an older guy.
Sometimes, instead of calling someone "old" outright, we try to be polite and say "he is 'older'" rather than "he is old."
"Older" is an adjective.
Elder, however, is only used to describe people only (not objects). It also has the sense of "older," but rather than just a difference in age, "elder" may imply seniority/rank as well. "Elder" can be either an adjective or a noun.
E.g.: The people consulted the village elders about this moral dilemma.
My elder sister received a PhD in linguistics. (Here, you can also say "my older sister received a PhD in linguistics").
The elders (or the elderly) are more susceptible to Alzheimer's Disease than the young.
The elder statesman came to settle the dispute between the junior members of the legislature.
And as Jmat said, we consider it rather rude to call the elders "old people." (That is because "elder" emphasizes seniority/rank, and "older" emphasizes age.)
I hope this was helpful.