Here's what we know happened at Hogwarts while Harry wasn't there, mostly based on what the DA members tell Harry when they meet again before the Battle of Hogwarts:
Snape becomes Headmaster, as announced in the Daily Prophet on the first of September. Alecto Carrow becomes the new Muggle Studies teacher, and her brother Amycus becomes the new Defense Against the Dark Arts teacher - both become Deputy Headmasters under Snape. Slughorn becomes the Head of Slytherin House and continued to teach Potions. Attendance at Hogwarts becomes mandatory for all appropriately-aged children who could prove their blood status. No Muggle-borns were able to attend the school, and only half-bloods with proof of their magical heritage are able to attend, which is why Dean Thomas, the son of a wizard murdered by Death Eaters after Dean's birth, is forced to go on the run.
Muggle Studies becomes a required class and educates students about how Muggles are "like animals, stupid and dirty" (as Neville tells Harry). The class teaches straight from the Ministry's anti-Muggle propoganda. Defense Against the Dark Arts is renamed Dark Arts. We don't know much about the curriculum, but we do know that the seventh-years are told to practice the Cruciatus on students with detention, and that the Killing Curse and Fiendfyre are covered as well.
The Carrows are also in charge of punishment of students, though most staff avoid reporting rule-breaking. The punishments for rule-breaking are cruel and varied, and include: the Cruciatus Curse, chaining students in the Great Hall, and physical corporal punishment with beatings and knives.
Dumbledore's Army continues to operate under the leadership of Neville, Ginny, and Luna. The DA rebel against the Death Eaters by refusing to practice the Dark Arts, standing up to the Carrows, and sneaking out to write rebellious graffiti on the walls. The trio of Neville, Ginny, and Luna attempt to steal Gryffindor's sword from Snape early in the fall, but are caught and given detention with Hagrid. Luna does not return to Hogwarts after Christmas, and Ginny leaves for Easter holidays and also does not return, so Neville eventually becomes the sole leader and suffers through a lot of torture for it. The open rebellion ends sometime in the spring when Michael Corner, a seventh-year Ravenclaw, is tortured relentlessly for sneaking out at night and freeing a first-year from the Carrows' chains. At one point in the spring, Neville, fearing for his life, flees to the Room of Requirement and remains there. The passage to the Hog's Head appears to Neville and the students when they get hungry, because the Room cannot produce food. The Room expands and creates hammocks to sleep in and washing facilities as more DA members come to live there - Seamus tells Harry upon Harry's arrival on May 1st that the DA had been living there for almost two weeks. Neville, Seamus Finnigan, Lavender Brown, Padma and Parvati Patil, Terry Boot, Anthony Goldstein, Michael Corner, and Ernie Macmillan are present and living in the Room when Harry arrives. The emblems of Gryffindor, Ravenclaw, and Hufflepluff are present in the Room, but not Slytherin, implying heavily that there are no Slytherin DA members.
There is already a great fanfic called Dumbledore's Army and the Year of Darkness which covers what happens at Hogwarts while Harry is away, but this is what the book expressly discusses as occurring during the year. Hope this helped!
