For example, if it's water damage on the tiles, you should fix the leak before attempting to fix the ceiling.
After that, what you do depends on what exactly is wrong with the old ceiling. If it's just nail holes from the 1x2's, you're talking about a little spackle and some paint.
If the existing ceiling is damaged by something like a leak, but the damaged area is small (about 3" in diameter), remove the old material, and you can construct a patch that is supported by the old material. If you happen to be this lucky, I can explain how to do that in another post.
If there was a leak, it probably damaged a larger area. You'll need to replace the damaged ceiling, and then get out the paint. To fix it, remove the drywall/plaster in the damaged area. Then remove additional drywall/plaster until you have a roughly rectangular area that ends about 1/2 way through the ceiling joists (the wood that forms the structure behind the drywall/plaster). Your goal is to create a seam between the old and new drywall and have both sides of the seam anchored to the structure of the house. If most of the ceiling is damaged, it may be easier to tear down all of it and start over. If the existing ceiling is plaster, consider ripping all of it off if 1/4 to 1/2 of it is damaged - it's harder to repair than drywall.
From there, you need to cut a new piece, or pieces, of drywall to fill the hole. If the existing ceiling is drywall, then it's probably 1/2" drywall and you can just use new 1/2" drywall to end up with a flat surface. If the existing ceiling is plaster, then it will be much thicker - probably 1". Since you need the new material to line up with the old, you'll need to use 1/4" drywall strips on the joists, followed by a sheet of 3/4" drywall to fill the area.
Next, you'll need to tape the joints with drywall tape. For a repair like this, I recommend the fiberglass mesh tape. It's much easier to use than paper tape, especially overhead - it's self-adhesive, while the paper tape is not. Pros like paper tape because it's faster to end up with a smooth final surface, but they have the practice to make it work well.
Then you'll need to smooth over the tape and any other irregularities with drywall compound. There's two kinds - water based or cementitious. Water-based is available already mixed in a tub, or as bags you mix with water. You will have to wait at least overnight for the water-based versions to dry and harden before sanding and applying the next coat.
Cementitious versions will advertise the drying time on the bag, and are only available as powder. You mix the powder with water, spread it on the ceiling, and then wait the time on the bag. Do not get the 5 minute version - that's only good for either very experienced people, or very small areas. You'll need to wait at least double the time on the bag before sanding - the time on the bag is only the working time. It is better to wait longer.
You'll be putting on 2-3 coats of drywall compound. So don't try to fill any gaps in your first application. The compound will be too thick, and not dry properly - it shrinks when it dries, and you'll be left with a more difficult problem to solve in later coats.
So apply a coat. Let it dry. Sand it mostly smooth. Then apply a second coat. Let it dry. Then sand it smooth (note the loss of "mostly" from the previous coat). Repeat until you have a flat-looking ceiling.
Finally, if all that seems like way too much work, you could have an acoustic ceiling sprayed on the old ceiling to cover the damaged stuff. It looks like cottage cheese. It only comes in white. If you paint it another color, it will be very annoying to remove in the future. And it's a really, really, really bad idea to try and apply it yourself - you really want to hire it out if you're going for that solution.
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