It means having a situation to deal with and being able to satisfactorily complete it. The person taking responsibility needs to be aware of the fact that they need to assume ownership of their own behavior and the consequences attached to it. It's important not to read this as an individual matter, responsibility involves others as well, especially when volunteering.
Being responsible is not to make things right, is not a matter of making mistakes or not. Responsibility refers to the self-awareness of our actions. It means that if we made a mistake, we should not hide it, we talk about it, we are accountable for it, and think about what can we do. It is reflecting that we can develop this skill.
When volunteers arrive we always emphasize responsibility. For example: If one day you don't feel well, let us now. Why? because it shows respect to the people on the project and not showing without previous notice can break people's trust and make them worry. In the same line, if you felt bad, didn't go to the project and didn't warn, don't be afraid to say so, it can happen. This is just a small example, but it's clear. Responsibility is not about achieving objectives and doing our job without mistakes, it is about being honest, understanding that our actions have an impact, and facing up to it.
Volunteering can help you develop this skill as you get a determined role, you have schedules, tasks, people that are counting on you being there. All these aspects challenge you to be responsible and in a friendly environment where you can feel free of talking and explaining.
To be a volunteer in itself is a responsible action, it is a form of responsible tourism, of knowing a place and really seeing beyond the tourist places, of knowing the people, the neighborhoods, the ways of life and not staying with a simple postcard.