Uncountables for abstract ideas usually become countable when qualified. responsbility [general] Being a manager, she has a lot of responsibility. [general - unqualified] Being a manager, she has a/the responsibility to run the department efficiently and a/the responsibility to follow the laws regarding working conditions. [qualified]

How many time Have you been responsible to do something for your kidnapped country ? Our responsibility is to be oppose of that flag !

People answered your question well, but I wanted to mention that many words follow the same pattern where they have an uncountable version and a countable version, so keep the information from the discussion in mind when you come across them.

Be aware that with the countable form of responsibility, there is another structure which often sounds more natural. Managers are responsible for running their department efficiently and complying with all labor laws. I’m responsible for the payroll. If your paycheck seems wrong, send me an email. A: I have to do the payroll every week. B: That’s a big responsibility. (It’s already been introduced. I’m expressing my opinion of it.)

If you use 'responsibility' as an uncountable noun it is considered a quality. "This job has less responsibility." "Responsibilities", however, is considered countable, because it is assumed that you could list them individually. They're both correct based on the context and what you really want to say with a slight difference in meaning. Good luck!