Adware, as the name suggests, is computer software that has the sole purpose of delivering advertisements: adware, advertisement, software. Sophisticated adware will match the advertisements they deliver to the browsing/computer habits of the host computer.
It is rare, and probably unique, that a computer user would request to view advertisements. Therefore, a proportion of adware installs itself on a users computer without their knowledge or permission. For adware such as this, the most common delivery method is to hide the adware program within legitimate software: browser toolbar plugins being a prime example. There are unethical software developers who profit/fund their project by hiding adware within their software; they may profit directly from the adware or be paid a lump sum by a third party source. This type of adware is also classified as malware and spyware, and can be removed by software applications like: Malwarebytes Anti-Malware, Spybot - Search & Destroy, and Ad-Aware.
Legitimate adware is integrated or inserted alongside an online service. The purpose of adware is to provide income for the software developers: this may be to increase profits or to cover the development and maintenance costs of a free service. The most obvious adware technique is to use a 'popup' or 'popunder' window that displays an advert. Popup's are used within software applications like Steam. Another example of adware is when a advert is randomly displayed by Youtube before a video is ran.