It's a little convoluted ...
Ohio State "National Championships"
2014 - Beat Oregon 42 -20
2002 - Beat Miami 31 - 24
1970 - Shared Championship with Nebraska & Texas (Although the Associated Press selected the undefeated Nebraska Cornhuskers as the national champion, the NFF and UPI polls were conducted before the Longhorns and Buckeyes played in their respective bowl games. The AP's was awarded after bowl season.)
1968 - Beat USC 27 - 16
1961 - Shared Championship with Alabama (The AP and UPI selected Bear Bryant's 11-0 Alabama team as national champions, with the Football Writers Association of America selecting the Buckeyes, who were 8-0-1. Although Ohio State was invited to play in the Rose Bowl, the school's faculty rejected the invitation 28-25, resulting in OSU students taking to the streets in protest. This was because OSU's contract with the Rose Bowl expired, which left the decision to the faculty. Minnesota ended up going instead, and beat UCLA 21-3.)
1957 - Shared Championship with Auburn (Although Auburn finished 10-0 and ranked No. 1 in the AP's final poll, the SEC placed the Tigers on probation. As a result, the Tigers couldn't play in a bowl game. Ohio State beat Oregon in the Rose Bowl, finishing 8-1 and ranked No. 1 by the UPI poll.)
1954 - Shared Championship with UCLA (Ohio State finished 10-0 and beat USC in the Rose Bowl to finish ranked No. 1 by the AP. But 9-0 UCLA was voted No. 1 overall by the UP and FWAA polls. The Bruins couldn't play in the Rose Bowl, despite winning the Pacific Coast Conference, thanks to the game's "no repeat rule." The Bruins played in the Rose Bowl in 1953, losing to Michigan State.)
1942 - No bowl game. The 9-1 Buckeyes won the Big 10 (5-1 conference record) and were ranked #1 by the AP at the end of the season.