Here is a general definition of a homeschool co-op: homeschooling parents rent a building with multiple rooms and teach groups of kids classes. An alternative is to hire professional teachers or subject matter experts to teach courses in their field of study or a general teacher tutor to teach multiple subjects. Usually more than one class is offered per age range at any given time and the subjects can vary greatly so the registration is a bit more like registering for college courses then what students in American public schools do, or at least those in middle and elementary grades. Many co-op's are one day a week but sometimes busy co-op's expand to two or three days a week.
You may ask how is a homeschool co-op unlike a very small private school? The answers I can provide based on my experience would be that the atmosphere is more home-y than any school I've ever been to. The kids want to be at the co-op, even if they are not thrilled with the content of perhaps one or two classes. In those cases sometimes the parent pulls the child out of the class. Other times the child chooses to take the "less than ideal" part so they can be there for the rest of the good parts of it.
Generally co-op's are optional for families which is not at all like any type of school; it somehow changes the whole dynamic and general "vibe" of it (hate to sound so touchy-feely but truly I can feel the energy and difference between a harmonious homeschool co-op and the tensions felt in other group activities we've pareticipated in with kids). Anyone who has ever felt they could "cut the tension in the room with a knife" cannot deny that there is another sense that humans have, a sense of "feeling" in the air. (This is not to say that people can't also get bad vibes from a co-op, it does happen, especially if problems arise.)
Some Reasons to Join a Homeschool Co-Op Instead of Doing 100% of Studies at Home with Mom
1. Some children can use some experience with a non-Mom teacher. It can be good to be held to a certain standard of studies and school work that comes from someone other than Mom. Perhaps this is most true for boys in upper middle school grades or for kids who have always been homeschooled who have no reference point for formal school and accuse Mom of expecting too much, being to strict, or other accusations that some homeschooled children make.
2. Sometimes another teacher is a better teacher of a certain subject. Other homeschool parents can be subject matter experts (such as former math teachers, former attorneys, and former engineers). Sometimes the parent just doesn't have the patience to teach a child a certain subject.
3. The other homeschool parent-teacher or the hired teacher may be passionate about a subject which inspires the student when their own parent lacks this passion and enthusiasm. I imagine this is more of an issue for high school content and some middle school grade subjects.
4. Homeschool Mom is burned out of teaching or trying to teach their child subject X and needs a break. Other teachers may bring a different technique, method, or have a higher energy level or passion that helps the student through their struggles or their plateau or past their resistance to learning the way Mom tried teaching them.
5. Homeschool Mom is burned out of homeschooling in general and needs a break. (Different co-op's have different arrangements and some are completely drop-off for some parents who are willing to pay higher prices for the privilege.)
6. It can be good for homeschooled kids to see what classroom learning is like, to learn classroom etiquette and some need or like to see how they compare to their peers in learning situations. (Some smart homeschool kids oddly assume they are dumber than their peers and develop poor self-esteem, this may be due to falsely assuming all learning should be easy and fun and when they struggle at home to learn, they assume all their peers must "get it" when that's just not true.)
7. The organization of papers and homework can help the student learn those skills. The way work is done at home doesn't always necessitate the organization of as many papers as formal classes do. It's good to learn these things especially if the student expects to attend some kind of school or college later on.
8. Homeschool co-op's can provide a different type of positive socialization. Kids can make some new friends than through other homeschooling activities they participate in. If a decent amount of time for lunch and recess is given then that's a great time for friendship making. Other friendships develop by being around the same kids all day for ten or more sessions per semester. Pre-existing friendships can become deeper. There is also the bonding that happens when kids go through things together including struggling over a homework assignment or other things that happen in a classroom.
9. Enrichment: some co-op's are heavier on fun or enriching activities which are fun and good. This can round out the child's experiences, providing different experiences that are taught at home.
10. Groups can hire teachers and split the cost yet still have a small student to teacher ratio. Ten kids taking a physics class from a teacher and splitting the cost is cheaper. For example, one of my son's teachers usually charges $110 an hour for 1:1 tutoring but in this case we split the cost ten ways.