How does a student fail to qualify for RAship, but fit the bill for TAship?

I wish to know how the Graduate Admissions Committees for top Universities in the USA decide about awarding Teaching Assistantship to a prospective student. If one had preferred both 'RAship' and 'TAship' in his/her application, does that mean he/she must first be eliminated from the RAship race and then awarded TAship as a consolation?

Also, if being a Research Assistant requires a stellar undergraduate research record, publications and so on, what are the factors which would have been considered for awarding the Teaching Assistantship?

To be very specific, if the student is from a country like India or China, would his English-speaking skills and Academic competence be reviewed?

Please help me understand how these two factors transform from mere selection criteria to those for awarding Financial Assistance.

The Universities, I received assistantship from are among the best in my field of interest in Computer Science.