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Malware is a catch-all term for unwanted and malicious software. This extends from 1980s viruses to modern day cryptolockers and adware.
Viruses are programs designed with the singular purpose of giving someone's computer, device or network a bad day. Traditionally they were created with Data Destruction or Denial of Service attacks on a local network in mind. They would often make themselves known to a victim with boot errors or screen takeovers.
Today they are much more subtle and try to hide themselves completely. They also exhibit much more subtle behaviour such as siphoning login credentials or performing reconnaissance for further attacks. Viruses today often have significant financial backing from either organised crime or nation states.
Popular infections include Cryptolockers (viruses which encrypt your files and demand a bitcoin ransom in exchange for the decryption key) and Remote Administration Tools (or RATs, which are commonly used to spy on victims through their own webcam).
In the late 90s/early 2000s widespread viruses would often make the nightly TV news, such as the ILOVEYOU virus. Today they are never mentioned.
Adware refers to malware which displays advertising to it's victim. These range from "friendlier" things like an on-by-default browser toolbar installation as part of an unrelated piece of software's installation (e.g. Google Search Bar in a WinZip install) to nastier varities which will hijack browsers and popup ads without any mention of them being installed.
Hardcore /g/entoomen and /tech/nicians consider any application which displays ads as adware (e.g. uTorrent).
Methods of Infection
Most malware infections rely on user PEBKAC. Whether it's an email attachment to the "funniest jokes ever.doc.exe.bat", malicious websites exploiting unpatched browsers or simple user inattention during software installation.
For the discerning /g/entooman, adware infection is more likely to occur because of more general security problems such as out of date software with known exploits.
Main article: Malware Removal.