Responsibility means to make and keep a commitment not only to yourself but to others.

[edit] Steps

Make small commitments at first. For example, promise to self that you will do your work without any one reminding you.
To develop personal responsibility, choose one goal and stick to it.
Keep time commitments. Turn in to work on time.
Another form of responsibility is admitting to a mistake, even if the consequences may seem severe. Responsible people admit their mistakes, learn from them, and go onto to accomplish other things.
Don't worry if you mess up. Just keep working at it. Responsibility is not something you gain overnight. But, if you just do what you're supposed to do, when you're supposed to do it, you will become responsible.
It takes practice and mistakes to become responsible. People learn from their mistakes and they become responsible by not committing the same mistake again.

[edit] Tips

Start simple. Make simple commitments and learn how to accomplish what was promised.
Make only promises or commitments that you can keep.
If needed, learn to organize daily tasks to keep promised commitments.
Make a checklist, a schedule, for each day of the week to stick to.
Know what your priorities are.

Just because you learn to be responsible doesn't mean you're immune to peer pressure. Avoid making friends who make foolish decisions. Surround yourself with friends who make well-thought out decisions and have shown themselves to be responsible.
Remember to have fun. Being responsible is not about taking everything too seriously.