To hand in is to turn in, to deliver, or submit something to someone. Usually, the term hand in means that one is going to submit the item by hand, or personally.
Before Sherry can leave for her winter holiday, she must hand in the research she has been working on to her supervisor.
The professor explained that if we were accustomed to handing things in late and made a habit of it, we might want to enrol on another course. He does not accept late work.
A hand out is an expression that means someone takes financial means from another in the form of charity. The expression handout usually has a negative connotation, as in: He was constantly looking out for handouts instead of working. A handout can also mean a paper meant to accompany a lecture or talk, given out by the speaker to help with his points.
David Cameron's advisors wanted to create a plan that they said would provide jobs and not handouts to Britons looking for work.
An assignment is a task or duty that is given to you, such as a work assignment, an academic assignment, or a military assignment that must be completed.
John will be finding out about his next assignment within the month. Then he and Julie will make a decision about moving, and whether to buy or rent a house.
Kate likes her boss very much, but her current assignment has her working in different branch of the bank, where the supervisors are unknown to her.
The term homework is usually used to describe assignments that are given to children to complete outside of school in order to reinforce concepts they are learning.
The teacher reminded the children that incomplete homework would result in a note being sent home to the parent.
Sadie made a rule in her house that only after homework is complete can the children play outside or ride their bicycles.