Translation:I see your teacher among the students.
This is called the "a personal". It is used in Spanish, but not English. You simply put the letter "a" in front of the direct object to say that the verb is directed at them. It's a really simple rule, but it is really simple and useless because when it is translated into English, the "a" is translated as nothing.
The direct object is the object that is directly receiving the action of the verb. I know, it's best not to include the word you are defining, but maybe explaining 'indirect object' well help. The indirect object receives the direct object.
I passed the ball. - ball is the direct object.
I passed Billy the ball. - again, ball is the direct object; it's what is passed. Billy is the indirect object, he is receiving the direct object. I am not passing Billy; I am passing (verb) the ball (the object being passed) to Billy (recipient of the DO).
arvind.par- All of these comments are on a page with a Spanish example. The question wasn't phrased in a way that indicated a misunderstanding of Spanish. It was phrased in an "I don't get that grammar concept" manner. This person appears to be an English speaker, that is why I explained a new concept to them in English.
In the example from Duolingo "Veo a tu maestra..." the verb is 'ver' (to see) so the direct object would be the answer to the question, "Who or what is seen?" The answer is "tu maestra" so that is the direct object. Since teachers tend to be people, the 'personal a' is necessary.
I became a bit confused with this. The personal "a" is present but I struggle to invision how "estudiantes" is a direct object here. Is "veo"/"I see" really an action? The only way I have been able to feel comfortable with it is using the method described here: http://www.englishlanguageguide.com/grammar/direct-object.asp "Your teacher was seen amongst the students by me" The sentence can be rephrased as a passive one, therefore the personal "a" is required. Hopefully that's correct!
This is a little accurate, but it's much more complicated. Both mirar and ver can at times be translated into English as to watch, to see, or to look. They usually aren't interchangeable in Spanish, and the English translation will not always be a direct equivalent. In other words, mirar is sometimes used in a Spanish sentence when the English sentence would use to see. Ver sometimes has to be translated as to watch in English. They have to make sense in translation instead of being a word for word substitution. Here's a more in depth explanation.
Because you have an "a" in there, its both incorrect and fairly nonsensical. The Spanish "a" in the original sentence is a "personal a" and has no translation in English. See http://www.studyspanish.com/lessons/persa.htm
Also it should be the possessive "your", not "you're" which is short for "you are". "Tu" = "your", "tú eres" = "you are" (informal).