Rounds in cryptography I need to make it clear I  know nothing about crypto so in that context I'm hoping to clear up some confusion:

As I understand it a "round" in a cipher is one encryption operation and a cipher like AES has multiple rounds. Did I get this right?

I've seen descriptions of attacks against a cipher say something like it "broke 4 out 8 rounds". In this case, if the plaintext wasn't recovered than how is it determined that any number of rounds were broken?

Thanks
Most encryption algorithm use a specific number of the same (or similar) group of operations. As example, AES consists of 4 different operations: SubBytes, ShiftRows, MixColumns and AddRoundKey. AES-128 (AES with a key of 128 bits) uses all this operations 10 times in 10 rounds. Well, not exactly. The last round doesn't have the MixColumns step. It would add nothing to the security which is why it's omitted.

If an attack has "broken 4 out of 8 rounds" it means that the same algorithm was broken if it would use only 4 instead of the normal 8 rounds. Using only 4 rounds instead of 10 in AES-128 is broken, even without special prequesits. Breaking a few rounds of a cipher is normal, because the secure properties of an algorithm have to slowly establish through the rounds. If 9 out of 10 rounds would be broken, that would be much more alarming. We can assume that the last round will be broken soon enough.

Now we have to mention one important thing: If an algorithm is broken it doesn't mean that someone can decrypt every message encrypted with this algorithm without the key. Nearly every attack on modern algorithms is not about this. There are different kinds of attacks: As example, a known-plaintext attack needs a special amount of plaintexts and corresponding ciphertexts to gain the key. Numbers of $2^{40} = 1\,099\,511\,627\,776$ are low, despite being something like 128 terabyte of encrypted data. A break with this small amount would be considered devastating for AES. For one special attack on AES-256 we still need ca. $2^{100}$ operations - too much in practice, despite being "broken" in the academic sense.