How you fix those cracks depends on the type of ceiling you have. Is it a smooth ceiling or a textured ceiling? The same products are used for each type of ceiling; however, the technique is quite different.
- Joint compound
- Mesh drywall tape
- Putty knife
- Safety glasses
- Sanding block
- Ladder or step stool
- Small piece of sheetrock
Prepare the Surface
If your ceilings are textured, you will need to scrape off the old joint compound to expose the crack. If the ceiling is smooth, you will be able to skip this step. Using your putty knife, scrape a 4-inch by 4-inch area the ceiling gently to remove the old joint compound. Once the joint compound is removed, score the sheetrock where the crack is located with your putty knife. The goal is to open the crack up so that you can press joint compound into the crack. You will want the crack approximately an 1/8 of an inch deep. Wipe the crack with a clean towel to remove any debris.
Apply several thin layers of joint compound to the crack using your putty knife, pressing firmly to help push the joint compound into the crack. Allow each coat to dry completely before applying the next coat. The coat should extend at least 4 inches beyond the edges of the crack. If you are working with smooth ceilings, spread the joint compound thinly along the outer borders of the 4 x 4 inch area. This will help you keep the surface smooth.
Once the joint compound is dry, lightly sand the area until smooth. Most professional sheetrock installers recommend using a wet sanding technique. To wet sand, dip the sanding block in water and squeeze out the excess water until the sanding block is just barely damp. Lightly sand the area to eliminate bumps.
Textured ceiling were very common in the last few decades; however, more and more people are choosing flat, smooth ceilings. A textured ceiling may have a design or it may simply be what is referred to as a popcorn ceiling. To repair a textured ceiling , you will need to create a pattern that mimics the rest of your ceiling. Practice this using a scrap piece of sheetrock. Follow the instructions from this video to learn how to texture a ceiling properly:
Smooth ceilings do not have any design in them. With this type of ceiling, you are finished after your joint compound has dried and you have sanded. With a smooth ceiling it is imperative that you smooth out all rough areas with a sanding block. Run your hand along the ceiling's surface feeling for any bumps or indentations. If there are any bumps, simply sand a little more until it feels completely smooth. For indentations, you will need to apply more joint compound to fill out the area. 
Finishing the Project
Because you have removed old ceiling joint compound and applied new joint compound, you will need to paint the ceiling. Choose a flat white ceiling paint and paint the entire ceiling with at least three coats of ceiling paint.