It depends on the variety of fig tree which colour they turn. We have brown turkey figs in our garden, they go a very deep green/slightly brownish colour when they are ready. Hubby is right, they practically fall of the tree into your hand and are soft to the touch. They will have the most delightful sweet smell to them as well. The flesh inside will be a brownish pink.
If your fig is the white marseille variety, they won't go such a dark colour, the flesh inside will be much paler as well, pink middle with white flesh around the outsides.
Here in the UK and in continental Europe, the figs are usually ready to pick around the end of August/beginning of September.
After this year's figs have finished, next year's will start developing really early, so make sure that you don't rub them off the tree thinking that they are undeveloped ones from this year.
Enjoy your figs