A DDoS attack is a specific sort of cyber crime. A Distributed-Denial-of-Service (DDoS) is a distributed Denial-of-Service (DoS). This occurs when a requested service is only to a certain extent or not at all available.
A zero-day vulnerability, in essence, is a flaw. It is an unknown exploit in the wild that exposes a vulnerability in software or hardware and can create complicated problems well before anyone realises something is wrong.