Something has happened! You spilled something, there’s a mark, or even worse; you ripped a bit. Well, fear not, as its ALWAYS savable.
I would start by saying though, that if you wash and iron your work, and store it properly, most of the below issues are fixed by washing it again.

Brown marks

This is a common issue with cross stitch that’s not been washed correctly. The oils from your fingers discolor in sunlight, making brown spots.
If you haven’t ironed the piece, you can just wash it in cold slightly soapy water for about 6 hours and you’ll be golden.
If you have ironed your work, then I’m afraid you’ll have to pick the stitches and redo them. There’s a quick guide on this at the end of the post.

Rust marks

Fairly uncommon on cross stitch, however, if you’ve stored it in the frame somewhere you may get rust.
Get ready for science! You want to find some Oxalic acid. You can find it online (it’s used by Beekeepers) or specialist cleaning stores. In short, it bonds with the iron of rust and makes a water-soluble solution. Mix a small amount (10g) with a tablespoon of water, and rub in. Wash it out under a tap after 15 mins and it should be as good as new!

Scorch marks

Happens ALL the time. This is most common in old samplers as they had candles around all the time.
Straight from the Smithsonian museum, the best way to clean them is bicarbonate of soda and water. Just rub it on and it will remove the spot by removing the topmost layer of threads.
Or you would use white vinegar (very small amounts) which will remove the burn color from the stain.

Blood

This usually happens during the sewing, following nose bleeds, however, the key here is speed. You want to be cleaning it the SECOND it hits thread/aida. The longer you leave it, the harder it gets to clean.
Salt. I would use a 3:1 salt to water ratio, and just rub it in. Once the salt has gone brown, wash it off. If the stain is still there, repeat.
In rare cases, the blood will have dried into the aida and you either can’t access it, or it won’t come out. In this situation get a bowl of warm water and add salt until it doesn’t dissolve anymore. Sit the piece in the water and let it sit. I would suggest covering it with something non-transparent to stop evaporation too.

Pen

Yeh, we’ve been there…
Cut a lemon open and drip it onto the pen mark. It should wipe straight off.
But limit the lemon juice, and make sure to give it a good clean afterward, as over time the lemon will bleach the threads.

Tea

As a stern tea lover, I’ve had this issue more than a few times. Coldwater works well, but if you have soda water/carbonated water that works best. Just pour it over, letting it bubble on the surface until its all out.

Thread bleed

This is actually fairly rare, normally as a result of incorrect washing, or a cleaning fluid.
This will take time. Be prepared.
The first step is a big flat pan. Lay it down face up, and add ice on top. Let it melt, add more. Repeat for at least 3 days, if not a week. The colder you keep the project the better, so you can keep it in the fridge or freezer if you want too. After this, wash it in cold water only. Dry it so its touch dry, but not perfectly dry. You need to iron it, stopping any bleed happening in the future.

Too late to save?

Sometimes it’s just too late. Maybe you only noticed something weeks after it happened, or the above cleaning doesn’t work. It’s OK, as there are ways around it.

Remove a stitch

If the area affected is small enough, then you can remove the stitch and put a new one in. This is actually quite common, and even if you need to remove a whole row, the effort it takes is often less than cleaning.
If the piece has been out in the sun a while, the threads can be a little bleached, and so sit the new thread in lemon juice for a few hours and it should match better.