1. Used to emphasize negation or absence, indicating that something is not the case and furthermore something else is also not the case
He didn't call, and he didn't even send a message
Ele não ligou, e sequer mandou uma mensagem
2. Used to express surprise or disappointment about a complete absence of something expected
The restaurant closed and didn't even offer an explanation
O restaurante fechou e sequer ofereceu uma explicação