What is the majority of this country's jail inmates?

The majority of inmates in county jails in the US are incarcerated.

The data in the following answer comes from the Department of Justice, Bureau of Statistics, Midyear 2012--the newest data available:You've made the question difficult to answer.

If you are asking the majority gender demographic for American jails, the majority is overwhelmingly male: 645,900 male to 98,600 female.

If you are asking what ethnic group makes up the largest percentage of the American jail population, then white (Caucasian) males is the answer you are seeking: 341,100, whites; 274,600, African-Americans; 112,700, Hispanics; 14,700, all other races; 1,500, mixed races (two or more).

If you are asking the age demographic, that gets a bit vague, but this is what is provided: 640,900, adult males; 98,100, adult females; 5,400 total juveniles; 4,600, juveniles held as adults; 900, juveniles held as minors.