The warm rent (Warmmiete) is your total monthly rent. It includes your cold rent (Kaltmiete) and utilities (Nebenkosten). It might not include electricity for your lights and appliances.
Cold rent (Kaltmiete) + utilities (Nebenkosten) = warm rent (Warmmiete)
Your warm rent is adjusted every year, because utility costs change based on your usage.
Your warm rent should be around 30% of your net income. In Berlin, the warm rent for an unfurnished apartment is €16 to €25 per m². Furnished apartments are more expensive.