“What else can go wrong today?”

You're having a bad day. You are at home with your husband. While you're writing a document on your computer, the program crashes and you lose all the work you've done. You complain:

What ELSE can go wrong today?

what else

"Else" means "other things". It's often used with questions:

Who else is coming?

We should definitely go. When else will we get a chance to visit Spain?

"What else" means "what other things". You start a question with "what else" when you have some examples of things, but you want more:

What else should we bring?

What else can I say? I've already apologized multiple times.

(something) goes wrong

When a plan, an action, or a situation "goes wrong", it means that there are problems:

What went wrong at your birthday party?

If anything goes wrong, call me immediately.

When an object, like a car or a toilet, has problems, you don't say that it "went wrong". Instead you say "it's messed up", "it broke", "it has problems", etc.:

The hood of my car is kind of messed up.

My laptop broke last week. I have to get a new one.

What else can go wrong?

When a lot of things have gone wrong in a day or a short period of time, and then you find out about another problem, you say "What else can go wrong?". This is a way of complaining about the number of problems. Because this is a type of complaining, it's best to use this phrase with people who you're close to.