I am currently writing a project report for my university project and I have the following question. In the introduction, I mention that in a following section I would explain my progress. My teacher corrected this usage of 'would' to 'will'. Which one is correct?
'Will' is correct. Because you intend to explain something. It's a pretty definite plan - something you are going to do.
Your 'will' is your intention - something you pretty solidly are going to do. If you used 'would' then it means you haven't yet made up your mind - and as the other answerer said - 'would' also means your decision is also conditional on another factor.
For example 'I would explain my progress - if I had time. I would explain my progress - if I had more data. I would explain my progress - if I thought you were interested.
So using 'would' we feel that you have not yet decided - so we are not, actually, expecting a progress update - and we are kind of waiting for you to tell us - what that conditional factor is.