If you require your sister to go to the party or you will tell your parents a secret about her, you oblige her to go to the party.
If you agree to go to the party when your sister asks, this is an example of when you oblige her.
If you are grateful to someone for giving you a gift, this is an example of when you are obliged.
I am obliged to you for your recent help.
When the power went out, we were obliged to fetch water with a bucket. The contract obliges you to meet the deadline.
The singer obliged with another song.
I am obliged to you for your gracious hospitality.
They obliged us by arriving early.
The soloist obliged with yet another encore.