Many things can damage a section of ceiling: humidity, age, and leaking water can all cause damage. Before you start to repair the ceiling, make certain you have addressed the underlying cause.
You will need a six-in-one tool, sheetrock compound or plaster, a trowel, mixing bowl, spatula, sandpaper, and paint. A shop vacuum will be helpful as well, for clean-up.
To start, scrape away any loose plaster or wallboard. Sand the area down once and apply an even coat of sheetrock compound so that the affected area is level with the rest of the ceiling. This may require multiple applications. After it dries, sand it and paint the area you repaired.