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I understand that the phrase "hacer a alguien hacer algo" means to make someone do something. My question is: is "alguien" the indirect or direct object? For example if someone were to say "I want to make him eat the food," would one say:
Quiero hacerlo comer la comida.
or
Quiero hacerle comer la comida.
Also, in either case, what part of speech is "comer la comida," or whatever someone is being made to do?
Both are correct. Per the DPD:
Los verbos hacer y dejar, cuando tienen sentido causativo, esto es, cuando significan, respectivamente, ‘obligar’ y ‘permitir’, siguen la misma estructura que los verbos de influencia: «verbo causativo + complemento de persona + verbo subordinado». Tanto hacer como dejar tienden a construirse con complemento directo si el verbo subordinado es intransitivo: «Él la hizo bajar a su estudio y le mostró el cuadro» (Aguilera Caricia [Méx. 1983]); «Lo dejé hablar» (Azuela Tamaño [Méx. 1973]); y tienden a construirse con complemento indirecto cuando el segundo verbo es transitivo: «Alguien lo ayudó a incorporarse, lo estimuló y hasta le hizo tomar café» (JmnzEmán Tramas [Ven. 1991]); «El alcaide de la cárcel le dejaba tocar el banjo todas las mañanas» (Cela Cristo [Esp. 1988]).
Notice that it only says "tienden" and therefore there is no single approach — you will get variety, although the most common is to use a direct object when the verb is intransitive and the indirect object is transitive. In your example, the verb is transitive (comer la comida) so the most common would be to use le.
Likewise, to answer your other question, per the DPD, the term is a "subordinate verb".
