The Dawes Rolls were created by the Dawes Commission, which sought to make a complete record of all members of the Five Civilized Tribes, as the Cherokee, Choctaw, Creek (Muscogee), Chickasaw and Seminole tribes were collectively known. The Dawes Commission was authorized in 1893 with the goal of breaking up the reservation system of the Five Tribes. (The Dawes Act, from 1887, had already started the process for other tribes in Indian Territory.) The Dawes Commission's early efforts at creating rolls were ineffective, so much so that the Curtis Act, passed in 1898, ordered new rolls be taken that completely would supersede the data previously collected. On April 26, 1906, Congress passed an act that closed the rolls on March 5, 1907 (although another 314 people were enrolled in 1914). The rolls are considered complete.
There are several Dawes-related resources online for those wishing to investigate their heritage. The Oklahoma Historical Society website has a database of the rolls searchable by first name, last name, tribe and index number.
For a potentially more rewarding search, visit the Archival Research Catalog at the National Archives website. There you can find scans of the entire Dawes Rolls as well as scans or summaries of many applications for inclusion.
Both sites also host records pertaining to the 1896 citizenship applications which were invalidated with the new rolls prescribed by the Curtis Act in 1898. While irrelevant to the question of whether an ancestor was on the Dawes Rolls, the documents might nonetheless prove to be a valuable genealogical resource. If you think you may have an ancestor who applied for tribal enrollment in the 1890s start with a simple search at the Oklahoma Historical Society. Be sure to read the disclaimer, which gives some details as to why these records were considered unsatisfactory.
The National Archives' “Native Americans: Highlighted Records” page also includes a search for 1896 Citizenship Applications.