C'est quoi vs Qu'est-ce que c'est

Bonjour Dominic, 

Both expressions mean "what is it/that?" but they sit at very different points on the formality scale.

Qu'est-ce que c'est ? is the standard, "correct" form you'd use in neutral or formal contexts — in writing, with people you don't know well, in professional settings, or in a classroom. It follows the full interrogative structure French grammar prescribes.

C'est quoi ? is colloquial and conversational.  You'd use it with friends, family, or in casual spoken French. It's extremely common in everyday speech — arguably more common than the formal version in informal contexts.

A rough English parallel: the difference is a bit like "What is that?" versus "That's what, exactly?" or even just "What's that?" said very casually — though the gap in French is actually wider, since c'est quoi would feel quite out of place in formal writing in a way that English "what's that?" wouldn't.

A few extra nuances worth knowing:

Qu'est-ce que c'est que ça/cela ? can sound slightly stiff or even pedantic in fast casual conversation among friends, whereas c'est quoi ? sounds completely natural. There's a middle ground too: C'est quoi ça ? adds ça for extra informality or mild exasperation ("What on earth is that?"). 

So the short version: qu'est-ce que c'est ? is the grammatically "proper" form; c'est quoi ? is what most French people actually say to each other on a daily basis.

Hope this helps!