Cuna: A crib or other small bed made specifically for a child or baby.
Tierna: This word is often translated as "tender" and is frequently used, as here, as an adjective of affection.
Heno: Hay.
Prestaban: The verb prestar most often means "to loan" or "to lend." However, it is often used, as here, to mean "to give."
Buey: Ox.
Bramaron: Bramar refers to the groaning sound of an animal. Here the verb is probably best translated as "bellowed."
Mas: Without the accent, mas typically means "but." The word isn't used much in everyday speech, where pero is generally preferred.
Sí: Sí most often means "yes." As can the English word, sí can also be used as a way of affirming or emphasizing what has been said.
Mírame: The verb mirar can mean simply "to look." In this context, however, it also carries the meaning of "to watch over." Mírame is a combination of two words, mira (watch over) and me (me). In Spanish it is common to attach object pronouns to the end of certain verb forms commands, gerunds (see amándome below) and infinitives.
Pensando en: In Spanish, the phrase for "to think about" is pensar en.
Me guardes a mí: This is a redundancy. In everyday speech, me guardes (watch over me) would be sufficient. Although in speech the addition of the grammatically unnecessary a mí might be done for reasons of emphasis, here it is used to help provide the right number of syllables for the music.
Amándome: This is a combination of two words, amando (loving) and me (me).
Da: In this context, da is the imperative (command) form of dar (to give) used when speaking to a friend or family member. This the phrase da tu bendición means "give your blessing" or simply "bless."
Haznos: Another combination of two words, haz (the imperative form of hacer, to make, used when speaking to a friend or family member), and nos (us). Thus the phrase haznos más dignos means "make us more worthy."
Mansión: Usually a dwelling place, but sometimes a mansion. In this context, tu gran mansión figuratively refers to heaven.

