You can use it in both formal and informal contexts. It's formal enough to use in a business setting, or even an academic paper, but it's also perfectly normal for casual conversation.

It's informal. I think it's actually a quotation--the caption of a 1941 cartoon. I'll explain that later. An engineering project often starts with a drawing. Before CAD (computer-aided design), the drawings were made on a special device called a "drawing board." Sometimes, if there's a small problem, you can fix it with a small adjustment. You can file it down and make it fit. You can put on another coat of paint. Sometimes you just need to throw everything away and start from the beginning again. In an engineering project, that would mean "going back to the drawing board." This is the famous cartoon I was talking about. It is by Peter Arno and it appears in "The New Yorker" in 1941. https://tinyurl.com/y6qe4eqb It is "black humor." There has been a terrible disaster. A new plane has just crashed. Apparently the engineers made a mistake. And the engineer responsible for it is acting as if it was nothing important, and saying cheerfully, "Well, back to the old drawing board."

"to start over again." Very commonly heard. To go back to the ole drawing board"