How should we handle exception in after triggers for both single as well as bulk records?

e.g. I am having a trigger on contact and I am updating Accounts in after triggers and exception occurs while updating Accounts then,
what is a best way of handling this exception (any type) in Salesforce?
1. Shall we show it using addError?
2. Shall we send an email to developer?
3. Only debug statements
Can I expect a small snippet of a code?