Question: Scientific literature has items called letters, communications and journal papers. They all seem quite similar in terms of format and content. What are the differences between them?

Answer: 
At my university in the USA, it would likely be illegal for the instructor or the TAs of the particular class to participate in this. There is just too much of a (perceived) conflict of interest since they are evaluating you for a grade for the class.

However, there would not be anything barring you from approaching a graduate student who was a **former** TA for the class; nor would there be a problem in approaching an instructor at **another** institution or someone who is at your institution who is **not** teaching your section of your class that term. You could even ask an upperclass student who took the class last year to tutor you.

Your department secretary may be able to connect you with a grad student who needs a few extra dollars and isn't in the conflict-of-interest position to tutor you directly. Or post something in your local craiglist or student bulletin board system.
The appropriate way to do it is explain what you want, and why you are willing to pay for it. You have to be careful to not say something like "I want to pass this course, and I'm willing to pay you to make that happen", since that would look like an offer of a bribe which would be illegal. It is not illegal to pay a person for personal tutoring. That is the point at which your concern should end. It is possible that university regulations forbid an employee from accepting such an offer, but it is the employee's responsibility to make that determination, not yours. If it is forbidden, or if the employee would simply feel uncomfortable with the appearance of conflict of interest, s/he can tell you so and may be able to point you to an alternative tutor.