"She looks at that mountain."
Translation:Ella mira esa montaña.
Well certain verbs have prepositions tied into their meaning, like buscar for example, which means to look for. You won't say buscar por because that would mean to look for for. Same thing here with mirar.
As for the use of "a", it can really vary. Right now, it would be helpful to remember to use it when people are involved with an action, for example:
Voy a ayudar a mis amigos = I'm going to help my friends. Voy a buscar a mi padre = I'm going to look for my dad.
That doesn't encompass the full meaning of the use of "a" in Spanish, but I hope this can at least help for now.
Certain verbs require a between them and a following infinitive. http://spanish.about.com/od/infinitives/a/verb_a_infiniti.htm
Other uses of the preposition a: http://spanish.about.com/od/prepositions/a/a.htm
Tener with preposition a: http://spanish.about.com/od/usingparticularverbs/a/tener_a.htm
More prepositions: http://spanish.about.com/od/prepositions/a/prepositions.htm
A good dictionary is always helpful: http://dictionary.reverso.net/spanish-english/a
"Monte" can mean either, but I think "montaña" is the most common for "mountain." This is just a problem with how DL auto corrects. They have identified "Ella mira ese" as being correct, then taken "monte" to complete a translation that is in their database. Had you written "Ella mira esa monte" then they probably would have corrected that to "Ella mira esa montaña."
It should not accept both here for this sentence. http://dictionary.reverso.net/spanish-english/mirar