There's two concepts here - "face" (literal translation) and "connections" (also a literal translation). Both are widely misunderstood.
AFAIK (and I'm not an expert):
Connections (guanxi) is fairly straightforward - it's all about who you know. However, there's a spectrum between casual acquaintances, people who do each other favors, extremely close friends, and family (in order of usefulness). It's often used as a bluff too (saying you know a guy, when it's all just a con).
"Face" (mian zi) is much more complicated. Since the term is used in English, it's easy to mistake what it really means. And it's also kind of vague - everything from "don't look undignified", to "don't let anyone disrespect you", to "have a good reputation".
Generally, Westerners think "face" means "be upfront, and willing to admit to an honest mistake". In China, this is rarely the case.
The unfortunate thing is that the reputation thing does not seem to extend to customer relationships. It boggles my mind how companies in China seem to be so willing to offer their customers substandard (or even unsafe) products for the sake of saving pennies here and there, especially in an age of weibo and capital punishment for the worst executives.