What is "malware", and what are its main types?

Malware = "malicious software": any program that performs harmful functions on a computer.

The word fuses malicious + software. It's an umbrella term covering, among others:

• Virus — attaches to a host file; needs the host to be run to spread
• Worm (Wurm) — self-replicating, runs on its own, spreads across networks without a host
• Trojan (Trojanisches Pferd) — disguised as something useful; sneaks in and spies/steals/alters data
• Ransomware, spyware, botnet clients, etc.

Virus vs. worm — the classic distinction: a virus needs a host and (usually) user action to propagate; a worm spreads autonomously over the network. Worms can therefore explode in scale far faster.

Tip: Basic hygiene from the source: don't open suspicious mails/links/attachments. Most malware still arrives via email and the web.