I recently got a job as a software intern at a big company. The intern program is loosely organized and basically was vaguely described a technical issue and told to implement a series of tests for it after explaining to him I am new to this and am not very experienced with any of this. I don't want to continuously say I have no idea what I am doing, but I literally don't even know where to start after he had explained a series of things to me way above my skill set. How can I professionally convey that I have no idea what I am doing and should I tell my manager how I am feeling? I got this job a week ago.
You are an intern. You are supposed to know nothing or very little about what you are doing. So, don't be afraid to ask questions.
However, you don't want to say you don't understand anything at all. Instead, you want to ask meaningful questions.
Your job is to implement a series of tests for a vaguely described technical issue. The first thing to do is to understand that technical issue. You start it by asking what it is. You want to be able to describe it in your own words. If there is some documentation, read it. If no, ask for it. Or ask someone to explain it for you. That some one can be your mentor. If you don't have one, ask them to assign one for you, formally or informally.
After you can describe the technical issue, you should have some understanding about it. Then you start to think about how to test it. Again, ask meaningful questions after you seriously think about how to test it. You may come up with naive ideas, impractical methods. They are okay as long as you have given them serious thoughts. Talk to your colleagues or mentors to see if they are workable. Receive suggestions and instructions from them and then correct your ideas. Once you have some workable ideas, start to implement them.
By the time you finish the implementation of the test and actually test them, your internship is probably over and you'll bring valuable experience home with you. Enjoy this wonderful journey.
Probably this is your first internship. You should not worry about your situation as it happens to most of the people. So be cool and you should have more patience. Okay, let us think about how to approach your manager for this. Before going to your manager, it would be good to do some homework on the tasks that were assigned to you.
First, list down all the tasks that were assigned to you. Then the most important thing is you should understand the problems/tasks.
Do some homework before going to ask for help: go through the wiki/docs related to the above problems at the company's websites (if there are any). If you understand the problems clearly then it is almost you have solutions for the problems. Then start one problem at a time -- start thinking about solutions for the problem that you think easy (the problem that you understand better).
If you are still confused after going through the above two steps, then it is better to directly go to your manager and request him to explain the problems/tasks again. As far as I know, managers can understand this and they are very happy to help you to be successful in your internship.
but I literally don't even know where to start
Then ask your mentor how to start.
You are not expected to understand everything at once, but you are expected to make progress. Therefore, learn things one by one, and let your mentor point out the things to need to learn about, along with suitable sources for that knowledge (books, people, ...).
If you have not been assigned a formal mentor, ask your manager about one, or (if the environment is informally structured) find one yourself.