A friend of mine asked this question and I wanted to post an answer here, because it's a common one for homeschoolers.
You may be aware that homeschooling is not the norm. As a result, people will have strong opinions about your decision to homeschool. These differences of opinion (from family and complete strangers!) begin the day your child is born, but since education is so important, concerns about homeschooling are often greater than whether or not you give your kid a pacifier or whether you feed on demand or on a schedule.
The best, brief answer you can give people is this, "This is the best decision for our family right now." I read this in one of Linda Dobson's books, I think, and have found it to be an excellent response. It's non-judgmental and straight-forward. But even armed with that response, I strongly encourage you to have concrete reasons why you're homeschooling. Then be prepared to give a simple, non-judgmental answer when people (or you children!) ask to know more about your choice. It's also very helpful to have those base-line convictions to fall back on during a bad day or bad week! For example, we began homeschooling because I enjoyed being involved in helping and watching Elizabeth learn and because Kindergarten was full day, and I didn't want to have E spend her best hours and so many of them away from home at such a young age. Now I've fallen in love with homeschooling, so we'll continue for additional reasons, but I don't need to go into all of that here.
One thing that's been driven home to me this year is that for every person that you please with your decisions, you're likely displeasing another. This realization helped me to calm down about worrying about what other people think, particularly about homeschooling, and just make the decision to do it. And not in a "stick-it-to-the-man" kind of way, but rather in a "stick-to-my-personal-convictions" manner. I've spent so much of my life trying to please everyone. It's freeing to realize that you can't!
Personally, I have not yet encountered too much opposition to homeschooling, other than the age-old question, "What about their socialization?" (I can address that in another post.) I live in Massachusetts, and up here it seems like whatever you do is cool, as long as you're passionate about it, but don't force it on anyone else. :) Our families have also been supportive, even if they do think we're crazy. (Love to you all!) I can imagine difficult situations, particularly trying to convince your parents that you're neither destroying their grandkids' lives nor judging that they sent you to school. In those relationships, I would say you need to extend a lot of loving-kindness. Have your reasons for homeschooling, and share them. If there's a good book that's influenced you to homeschool, lend it to them so they can get an idea of why you're excited about it. And pray through it all that you would grow in grace as you navigate the relationship. You may never convince them that homeschooling is a good alternative to pubic or private school, but at least they'll know where you're coming from and that you're making an informed decision.
We've only been homeschooling for eight months. I am not an authority, but I hope this is helpful!
More seasoned homeschoolers, would you like to add anything? Please do so in the comments.