Any software designed to do something that the user would not wish it to do, hasn't asked it to do, and often has no knowledge of until it's too late. Types of malware include backdoor, virus, worm, Trojan horse.
Malware typically affects the system on which it is run, e.g. by deleting or corrupting files on the local disks. Since Internet connections became common, malware has increasingly targeted remote systems. An early example was malware consisting of a malicious e-mail attachment that targeted security flaws in Microsoft Outlook (the most common e-mail client) to send itself to all the user's contacts. A more recent kind of malware "recruits" the infected computer to become part of a botnet consisting of thousands of infected computers that can then be remotely controlled and used to launch DDoS attacks.