The production shows are typically a revue of some sort, based on a musical theme such as movie musicals, or Broadway hit shows, or rock and roll through the years. They will do song and dance numbers to several tunes with set changes for each. The norm for the past several years has been to have just two lead singers who are miked - one male and one female, with the others doing background vocals. The shows are done to a high standard.
You might also have a featured performer such as a 'headliner' comedian or singer. Note that these are rarely 'big' names anymore, but more likely those who make the rounds on many cruise ships and lines, and also do Las Vegas shows. But still, they are quality acts.
Production shows will usually be done twice on their scheduled night(s), with a show for those with early seating dinner, and then one for second seating dinner. That way everyone has a chance to see it. You will not see production shows every night of the cruise. For a 7-night cruise they will normally do two different productions on two separate nights.