Design and Production is when you have a text that everyone is happy with, you start making it into a book. The page size will be decided (you are unlikely to have much say in this). Then someone will mark up the type, which means that they will choose the typeface (font) to be used, the type size, the number and length of lines on a page, what the chapter openers will look like and a myriad of other things of this kind.
The book will now be typeset and in due course you will be sent page proofs to read. Proofreading is when your editor should warn you when proofs are due and when you need to return them. This is not an invitation to rewrite, just to get it right. The publisher will employ a professional proofreader to do a final check, so in addition to any spelling or grammar mistakes that may have slipped through you should be looking out for things that only you will notice, such as errors of fact.
Mark any corrections on the pages you have been given in something bold like black felt tip or red pen. Timid pencil marks are easy to overlook. Write your corrections clearly, accurately and completely. Don’t leave the person who is typing in your amendments to guess at what you mean or to finish your sentences for you.