There's FanFiction or FimFiction (if it's set in someone else's work) or FictionPress if it's an original piece. Wattpad and a personal blog have been mentioned, but there are also the collection of NaNoWriMo (National November Writer's Month) sites out there, and plenty of other sites available.
I, personally, use FanFiction to sharpen my writing skills and have people read my work. It's a very active site, and there are plenty of writers there willing to offer feedback. There is also a 'beta finder' of sorts, but I personally wouldn't use it (most never update their information, or simply never get back to you).
I've had quite a few writers trying to entice me to join them in the NaNoWriMo competitions, and as I understand it there are some groups that are super active and supportive within that circle. I have no personal experience there, so I will not act as if I do.
There's also Google Docs, where you can place the story online and share a link with whomever you want. I'm sure there are other more private options as well, but none I can think of off the top of my head.
If that isn't what you're looking for, you might want to google around, see what other options there are. This is the internet, there's always another way.