that they are asking you to provide contact numbers for your references so they can contact and get a reference.

Is what they want.

So basically if you are in with org A, you'd give that orgs number. Generally bookers want at least two references, and in some cases three.  

Generally they don't care about reference names that much, because they'll verify with bookers and not girls. And all orgs know each other and know each other's numbers. It's a tightly knit system.

is to use the format in the link below.  It has worked for hundreds of guys here.