In the UK, people who work with no contract are called "casual workers". They can leave without giving any notice but, by the same token, the company can fire them on the spot. However, I believe that if an employee thinks that they have been fired unfairly or for spurious reasons, they might be able to take the company to court for unfair dismissal. In the first three months of a job, you can be fired for no reason whatsoever!
Many companies do have a contractual arrangement with their employees which lays out the rights and responsibilities of both parties. In the Civil Service, the agreement was one months' notice on either side, so when I resigned I had to work for one more month after handing in my resignation. By the same token, if they wanted to dismiss me, unless it was for professional misconduct, they had to let me work for another four weeks after telling me I was fired.
There are many different terms for people who work with a contract (fixed-term, permanent etc).