One, I think it means that you have something that is your job to get done. This is why my kids have had chores since they were 2. They evolve and they get minuscule amounts of money for doing some of them, but they are theirs to do. If they don't do them on their own, the kids are talked to about the consequences of not doing the job.
Second, I think responsibility means that when you do something wrong you own up to it and do your best to fix it. This week we put this into practice by having Little Man write an apology note to his teacher for misbehaving one day. We also talk about consequences here and include our kids in their consequence choosing.
Many, many ideas are out there for teaching responsibility so take some time to look them up and this week Be Responsibile.