what exactly is the binder?

Does it just go around you.. is it put on like underwear?

Mine is a one that wraps around you and Velcro's closed.

Is the velcro kind bulky under clothes?  Can you see it?

I had a thigh lift and I have a below the knee girdle like binder, it has zippers on both sides from the waist to about mid thigh.  I love it, but gravity does pull it down.  It is also crotchless (TMI I know)

I didn't like my binder under clothes.  It offers great support, but it does get bunched up.  I was constantly having to tug it back into place which was sometimes very painful.  I didn't mind wearing it underneath a loose fitting dress, but couldn't wear it with anything else really.  Once I started wearing things other than sweats (I'm still not in "normal" clothing), I switched to the Spanx/Assets/etc.  

Typically, will a surgeon supply a binder or do you need to purchase this independently?

I think it varies by surgeon...I had to buy my own, but they had me get it from a site that also sells to surgeons who give them to patients.  They charge I am sure, but some will provide them at the time of surgery.