This is an inaugural medal, but I am not sure of which type. From "Collecting Political Americana" by Edmund B. Sullivan (1980), Inaugural medals were commissioned by an inaugural medal committee, and were first issued in 1896. The so called "Official" issues were sanctioned by the national party organization with the president-elect's approval. A subcommittee would oversee the submission of the sculptor's designs, the final choice, its production, and its distribution. Often, the nation's top sculptors submitted designs. Augustus Saint-Gaudens, well known numismatically for his magnificent reverses on the eagle and double eagle gold pieces of the period, was selected at Theodore Roosevelt's urging to commemorate the 1905 inauguration. (There are two official issues for this inauguration.) On this medal, known as the Tiffany Roosevelt (because the famous New York firm handled the production), the ebullient Teddy is portrayed as a rather stiff-necked Prussian Junker. Although not one of Saint-Gaudens better efforts, this medal is considered the most desirable in the entire series because of its low mintage and the artistic status of its creator. Note: Do not confuse these "official" medals with the United States Mint's Presidential Series, which are usually issued in an unlimited amount since 1865 for each inauguration, and available for only a few dollars. It is recommended to read the well illustrated studies by Julian, Dusterberg, and MacNeil. (My reference book has neither photos nor descriptions for these 1905 issues.) If you are fortunate enough to have one of the "official" medals, it could be quite valuable, and certainly merits a bit more research to determine if it is one of the U.S. Mint's Presidential Series or not. If it is an "Official" medal, here are some values mentioned, keeping in mind they are from 1980. Firstly, the scarcity of medals issued prior to 1950 make them extremely attractive to collectors. The pre-1950 medals seldom had production of more than 3000 bronze pieces, and sometimes as few as 75 pieces (such as the 1924 Coolidge issue). Prices range from $50 or less for modern bronze issues to $1000 or more for the 1929 bronze Hoover, 1933 bronze Roosevelt, and the 1953 silver Eisenhower; $4000 for the 1917 Wilson, $5000+ for the 1904 (typo in book? called it a 1905 elsewhere) Tiffany Roosevelt and $15,000 for a gold Taft issue. These medals were not originally intended for distribution to the public, but only for circulation among inaugural committee members and other selected party members. I wish it were possible for me to give you an educated opinion as to whether your medal is worth a few dollars or thousands of dollars, but I can not. At best, I can only alert you to the slim possibility that it may have considerable value. If nothing else, I have learned a few things by reading about this facet of exonumia of which I had virtually no prior knowledge, and I hope you found this information interesting as well.
