The CRC32 found in zip and a lot of other places uses the polynomial 0x04C11DB7; its reversed form 0xEDB88320 is perhaps better known, being often found in little-endian implementations.
CRC32C uses a different polynomial (0x1EDC6F41, reversed 0x82F63B78) but otherwise the computation is the same. The results are different, naturally. This is also known as the Castagnoli CRC32 and most conspicuously found in newer Intel CPUs which can compute a full 32-bit CRC step in 3 cycles. That is the reason why the CRC32C is becoming more popular, since it allows advanced implementations that effectively process one 32-bit word per cycle despite the three-cycle latency (by processing 3 streams of data in parallel and using linear algebra to combine the results).