What does family mean to me? That's a really tough question to answer. I have a large family and we have always been close. They are everything to me- support, love, reality check, celebrations, sadness, hurt and pure joy.
I have life-long friends who meet that criteria as well and are considered family. My son calls them "aunt and uncle" and we usually refer to them as cousins because it's more true than saying "my best friend since Kindergarten".
I truly think that when you have a loving and close family you take it for granted. I know that I always have. I never really thought about how lucky I am to have parents, sisters, nieces, brothers-in-law, aunts, uncles, grandparents, great grandparents, cousins and so many others in my life actively.
After marrying my husband, I stopped taking that for granted. His family is not like this. They don't communicate, they aren't generally supportive of one another and there is not really a level of comfort when we're all together. He loves them, but can and will go weeks/months without hearing from his siblings and it is usually my husband who will pick up the phone or send a "is everything ok?" email. It's sad, but it's their reality. Given his choice, my husband prefers to be with my family. They love him like their own and for the first time he has seen what it's like to be loved, supported and celebrated.
So, to answer your question about what is important about family... they are there no matter what. Good times and bad, sickness and health. When my FIL was killed in an accident my parents got on a plane that night and my grandparents drove 4 hours just to be with us. My aunts, uncles and cousins who were within driving distance were all at the wake and funeral. Those who couldn't attend called, sent flowers and cards. They were all just there... we didn't have to ask, they just knew we needed to be loved.