"Sometimes my friend and I study on Tuesdays."
Translation:A veces mi amigo y yo estudiamos los martes.
It's simply a difference between the two languages. While in English we say "on" a particular day, in Spanish a definite article is used instead. When translating from Spanish to English, then, we turn the article into "on" to make the sentence make sense. I imagine Spanish-speaking English learners experience the same difficulty: "Why is there no 'the'? Where is this 'on' coming from?"
For practical purposes, we can consider that under most circumstances a definite article preceding the name of a day of the week can be translated to "on." "I go on Sunday." "Voy el domingo."
The exception to the need for an article involves many instances following the verb "ser." "Today is Sunday." "Hoy es domingo." This exception does not hold up, however, where the English sentence would say "is on" rather than simply "is." "The concert is on Sunday." "El concierto es el domingo."
In English, I could say that "I go bowling Wednesdays." English is less consistent, but the fact that it can be done in some cases without a preposition shows that it's not needed for comprehension. Spanish doesn't happen to use a preposition here, and English sometimes does. There's no reason why Spanish should use one. It's just the way it is.
Amiga worked for me. Perhaps the error was elsewhere in your sentence. If you post your whole answer we can help figure it out. It can be confusing because Duolingo doesn't always tell you where your mistake lies, and it doesn't give all the possible correct iterations. So when you see that the 'correct' sentence Duo gives has "amigo" and you had put "amiga", it's be tempting to think that this is why you were dinged. It's possible that amiga wasn't accepted in the past and was updated before I answered the question. Hard to say unless we see your whole answer :)
When I hover over "study" the pop-up shows...(nosotros) estudiamos. Since I didn't see nosotros anywhere I estudian which was marked wrong, I'm guessing since the sentence mentions a "friend and I" that I should have realized that "friend and I" become "we" becoming an invisible "nosotros."
The word "a veces" can go at the beginning (a veces my amigo y yo estudiamos los martes") to mean "sometimes, my friend and I study on Tuesdays (direct translation)". However, it can also go in the middle of the sentence, and be written as: "mi amigo y yo a veces estudiamos los martes," meaning "my friend and I sometimes study on Tuesdays"
Note that the speaker is included in the compound subject 'mi amigo y yo'. This makes it grammatically first person, and plural. If you were to replace 'mi amigo y yo' with the appropriate subject pronoun it would be nosotros.
Hence the correct verb form is estudiamos (1st person plural), not estudian (3rd person plural).
If the subject was, for example, 'mi amigo y su hermano', this compound subject would be third person plural (the replacement pronoun would be ellos) and, in that case, estudian would be the verb form to use.
The distinction is not so obvious in the English because study is used for all but the 3rd person singular form. But, if you replace 'my friend and I' with the corresponding pronoun, it would be we i.e. first person plural.