Without a doubt, the most common image format used by digital cameras and other image capture devices is called "JPEG." The name "JPEG" stands for Joint Photographic Experts Group, the name of the committee that created the standard. The group was organized in 1986, issuing a standard in 1992, which was approved in 1994 as ISO 10918-1. JPEG is distinct from MPEG (Moving Picture Experts Group), which produces compression schemes for video. An older and also popular format is called TIFF. Originally created by the company Aldus for use with what was then called "desktop publishing", the TIFF format is widely supported by image-manipulation applications, by publishing and page layout applications, by scanning, faxing, word processing, optical character recognition and other applications. Adobe Systems, which acquired Aldus, now holds the copyright to the TIFF specification. TIFF has not had a major update since 1992. Wikipedia.
To compare the two formats and decide which is the best for preserving digital images, if either, one of the best resources is the Still Image Working Group. This group is involved in a cooperative effort to develop common digitization guidelines for still image materials (such as textual content, maps, photographic prints and negatives). The expectation is that this work will enhance the exchange of research results and developments, encourage collaborative digitization practices and projects among federal agencies and institutions and provide the public with a product of uniform quality. It will also serve to set uniform quality and establish a common set of benchmarks for digitization service providers and manufacturers.
The current status of the standards is available in draft form listing links to the organizations that develop and maintain digital imaging standards. A complete summary of the formats is available from the Library of Congress in the site Sustainability of Digital Formats, Planning for Library of Congress Collections.
The not so simple answer to the title question of this post is contained in a chart in the Sustainability of Digital Formats site. The clear preference is for TIFF images, but JPEG images are still acceptable. Note that RAW images are designated as less acceptable.