To me, work is what you do to make money. It doesn't necessarily mean something you like, something that you studied for or even something that makes you a lot of money. Work can mean anything from mowing lawns and babysitting to being a lawyer or CEO. Work is what pays the bills. I worked as a grocery store cashier. I made barely more than Wisconsin minimum wage, worked less than full time and it sucked and I hated it, but it put gas in my car and funded the fun things I did
A career is what you've studied for and have plans to do for a much longer time. A career means you know you're good at it and it something you are successful with. Careers generally require some form of education. Whether it be studying air conditioning or getting your masters or PhD. I want to go into education as a career, but I want to get a masters in school administration and a degree in political science and work for the board of education. The difference between this and work is that where I worked I didn't have a goal or real possibility for promotion. I was briefly trained but never had to study for it. With my career I have goals, a plan to move ahead and back ups. I know it's something I will be doing for the long haul.
A vocation is something you are called to do. I always associate this with God and religion. I've always heard it used in that context. But it really just means your purpose. This is not a job or work or career, but why you're on this planet and how you're meant to make your impact. I like to think that I am here to make a difference in the American education system. That is my purpose and what I believe I was called to do. Your vocation can factor into your career or work or it can not. My vocation factors into my chosen career, however, a man who works as a manager at a company may find his vocation is to inspire people through art he does in his free time.