If you have a situation where several people gave assignments-all due very quickly-how would you handle the problem?

This situation shows that there is no unity of command as the assignments are given to the assignee by several assignors or bosses and not one single boss. Anyway, you should prioritize your work, You should handle what is important and urgent first. You can never handle the assignments simultaneously. You should let the assignors know that you have other assignments, and you are going to handle them based on their importance and urgency. Do not overpromise and underdeliver. Manage your time well. Avoid time wasters. If an assignor tells you that he needs his assignment to be completed quickly, clarify how quickly. Ask him to be specific. If you feel that you can not meet the assignor's deadline because you have something else more important and more urgent, let him know this before you accept the assignment. Communicate and clarify before you carry on with the assignments.

I can understand the situation, it creates lots of pressure as well as tension too. But by maintaining proper cool and sorting the work as per urgency, and completing part by part can solve the issue.

In every process there is a head that will make sure that a particular task assigned to a process was being met. Once you've got tasks beyond the process you are handling it portrays how weak the quality system of the company. But anyhow as the question states two powerful skills must take place, time management and prioritizing. A person must know how to manage his/her time effectively thus, prioritizing comes along. You need to weigh the importance or criticality of the task to the output of the work. If there would be delays that might happen feel free to tell your superior about it. Don't ever commit to something if you know it is close to impossibility. But also decide if you really can't handle the job. Remember, delays and objectives not met shows also your credibilty. Proper decision making and proper prioritizing.

I wont be able to do everything at the same time but would need to organize things to know what is the most important project needs to be done first and at the end of the day you will realize that you have accomplished what was asked to be done.

Find out which one needs to be done first. You can always ask your boss to help prioritize them. Then conquer them one at a time

That is the time to show your ability how much hard working you are give your100 percent ....

Depends upon the task or assignment importance.

Time management can resolve this issue .

Set priority based on urgency and hierarchical positions, and communicate your execution plan to everyone, if anyone requires changes, it have to be officially stated through a memo CC all to avoid backlash in the future

This is where we make mistakes without concentrating. Therefore, we need to give preferance to important work first.  

Delegation to right people and complete them as per priority