It is different for every family that does it, and often even for every individual child. As stated above, some people do online school which means you have teachers you corespond with over the internet. The online school sends you your materials and you complete the work and get a grade, just like regular school, only done on the computer from home or the library or the road if you travel a lot.
Some people homeschool through a private school or school of corespondance that sends them their books and materials for assignments, labs, projects, etc through the mail. With this type of home schooling, you usually send a report to the base school at set intervals.
Some people attend charter schools that let them go in to school once or twice a week, collect work, turn in work, and get help if they need it, though this isn't really what most consider to be homeschooling. Neither is online school to some people.
Some people take full responsibility for their own/their child's education by doing more traditional homeschooling. The parents and children choose their own curriculumk, text books, work books, lab equipment, videos, software, and any other materials they wish to use. They also choose their own subjects based on the child's individual interests and goals, for example, if the child wishes to go on to university, they will study the important core subjects that universities look for on transcripts as well as a variety of electives based on individual interests, needs, and goals. I personally am interested in robotics, animation, creative writing, and music, so this year I've been studying Programming and Robotics with various books and robotics/electronics kits for labs, 3D Animation with computer software, and creative writing with a textbook and books frokm the library. I also takek music lessons in the community and I am a member of a band. In this form of homeschooling, the state may require that you take a yearly standardized test to show that you are performing at or above grade level. In the states that are a bit more strict about homeschooling, your parents may need to send in a quarterly report, just like a report card in a way. Your parents may also give you tests at home if you all agree that this is what will be best for you, but it isn't required.You may also get grades like you do in school, or you may be on a Pass/Fail system, or you may have a totally unique grading system, or no grades at all. I take tests, but not for grades. When I take a test in a subject it is only to determine how far along I've come in that subject. The "grade" is never final because I homeschool to learn, not to just get by with a passing grade. If, for example, I took a test on a chapter I did in Algebra and I didn't do well on a few questions, instead of just going on to the next chapter, I go back and review the stuff I didn't get right on the test and then take the test again before moving on. For me, all tests are for is to tell me when I can move on to something more challenging. For me, and many others, tests aren't always in the same format as they are in public school. Sometimes in certain subjects, I'm tested by being given an extensive project or presentation to do to show my understanding of the topic cover, or asked to write a paper on it, or give a mock lesson on it as if I were the teacher. You know you truly understand something when you can accurately and confidently teach it to someone else. Often my mother, adult friends, or friends closer to my age who are in college or who have already mastered te subject will act as audience durring presentations or re-teaching activities, checking me if I don't seem to understand something. I only do this for certain subjects and topics though.
Homeschoolers sometimes do a lot of the things you may do in public school, but sometimes they do more as well. Many homeschoolers do a lot of hands-on activities like labs in science. This year, I am taking Chemistry, and I have all the same lab equipment you would use in a public high school chemistry class, just on a smaller scale, and with lab instructions specific to homeschooling (not requiring large groups of students, or very-hard-to-obtain chemicals). It was the same in Biology and in Physical Science. I have homeschooled friends who take thier science classes in a co-op group (kind of a homeschool class where parents and community volunteers act as teachers for each subject), and still other homeschool friends who take all of their highschool science classes (and some other classes) at the local community college as duel-enrollment students (just like regular highschool duel enrollment).
Homeschoolers don't miss out on the social aspects of school either. They certainly do not spend all of their time inside the house. Homeschoolers spend a lot of their time is spent out in the community, learning and experiencing life in the real world instead of in a house or in a classroom alone. Many homeschoolers take classes offered in the community such as art classes at an art studio or museum, musical instrument or voice lessons as well as band or chorus classes for homeschoolers at local music stores or schools, fencing lessons, swimming lessons, horseback riding lessons, classes and programs offered through local childrens museums, science museums, or history museums, classes or programs offered through the library, community/youth center, YMCA, or other Parks and Rec programs, dance class, and so on. Naturally, these are great opportnities for homeschoolers to interact with others of all ages, homeschoolers and public schoolers alike. There are also clubs outside of the regular public schools such as riding clubs, clubs offered through libraries and community centers, drama clubs at local theaters, boy scouts, girl scouts, Boys and Girls Clubs, youth group for those who are into the church scene, OM, academic teams, community sports, individual sports like martial arts, tennis, fencing, swimming, etc, and much much more. Many cities or counties also have homeschool organizations or co-ops where, as stated above, homeschoolers can take classes with other homeschoolers as well as go on frequent feild trips (though any homeschool parent or group of parents can arrange a feild trip when they want to), join clubs sponsored by the group (as many as can be thought of and started by the members), work on a yearbook or newspaper staff, attend dances or holiday parties (most groups have something along the lines of a Not-Back-To-School party each fall where the homeschoolers may go to a theme park or some other sort of outing) volunteer service projects, prom and graduation ceremonies for older students, outings on weekends, park days where younger kids meet up to spend an afternoon playing and pic-nicking in the park (uwsually weekly or monthly) and so on. Some even participate in private school sports competitions as well as regional and state science fairs and spelling/geography bee. There is no lack of social interaction.
The last form of homeschooling I can think of is unschooling. In this kind of home education, there is no structured school day, no tests (unless the child wants them), no grades. All of the learning and "school work" is directed by the child. This is usually best when started at an early age so that the child's natural desire to learn remains in tact and the child doesn't just become lazy or unmotivated. I am what I like to call a homeschool/unschooler because my schooling is all self-directed, but I still choose to do the book work and testing along with the creative hands-on stuff because I feel it is in my own, individual best interest.
I hope this answers your questions