A model release, known in similar contexts as a liability waiver, is a legal release typically signed by the subject of a photograph granting permission to publish the photograph in one form or another. The legal rights of the signatories in reference to the material is thereafter subject to the allowances and restrictions stated in the release, and also possibly in exchange for compensation paid to the photographed.
Publishing an identifiable photo of a person without a model release signed by that person can result in civil liability for whoever publishes the photograph.
Note that the photographer is typically not the publisher of the photograph, but usually licenses the photograph to someone else to publish. Liability rests solely with the publisher, except under special conditions. It is typical for the photographer to obtain the model release because he is merely present at the time and can get it, but also because it gives him more opportunity to licence the photograph later to a party who wishes to publish it. Unless a photo is actually published, the need (or use) of a model release is undefined. And, since some forms of publication typically do not require a model release (e.g., news articles and other editorial use), the existence (or non-existence) of a release is irrelevant.
Note that the issue of model release forms and liability waivers is a legal area related to privacy and is separate from copyright. Also, the need for model releases pertains to public use of the photos: i.e., publishing them, commercially or not. The act of taking a photo of someone in a public setting without a model release, or of viewing or non-commercially showing such a photo in private, generally does not create legal exposure, at least in the United States.
The legal issues surrounding model releases are complex and vary by jurisdiction. Although the risk to photographers is virtually nil (so long as proper disclosures of the existence of a release, and its content is made to whoever licenses the photo for publication), the business need for having releases rises substantially if the main source of income from the photographer's work lies within industries that would require them (such as advertising). In short, photo journalists almost never need to obtain model releases for images they shoot for (or sell to) news or qualified editorial publications.
Photographers who also publish images need releases to protect themselves, but there is a distinction between making an image available for sale (even via a website), which is not considered publication in a form that would require a release, and the use of the same image to promote a product or service in a way that would require a release. Whether or not publishing a photo via the internet requires a release is currently being debated in the 9th Circuit Court of Appeals. It is likely that any and all exposure to the public of unreleased photos via any vehicle will constitute civil liability for the photographer.