I can't believe (someone) would (do something)!

When someone who you like and respect does something bad that surprises you, you use this phrase. For example, if your husband yells and says something mean to your sister, you would tell him:

I can't believe you would say that to her!

In that situation, you can also say:

I can't believe you said that!

But there is a small difference between the two. "I can't believe you would ___" sounds more surprised and hurt.

You can also use "I can't believe you (did something)" for a surprising positive action. For example, if your wife bought you a really amazing and expensive birthday gift, you could say:

I can't believe you got me a motorcycle!

If you said "I can't believe you would get me a motorcycle", it would definitely sound like you were angry at her.

such a thing

"Such a thing" means "a thing like that". It's mostly used in two totally different situations. One is in sentences like the one at top. Another example:

How could you do such a thing?

This is a sentence that you say when someone does something horrible to you.

The other situation where "such a thing" is used is in the expression "There is such a thing as ___":

Is there such a thing as love at first sight?

This means "Does love at first sight exist?" or "Is it real?"

The negative version of this is "there's no such thing". Notice that it doesn't include "a" before "thing":

I remember being really disappointed when I found out there was no such thing as Santa Claus.