Gaumukh is the end (terminus) of Gangotri glacier, which is the birth place of Bhagirathi river.
To get to Gaumukh, one needs to do an 18km (supposedly intense) hike from the Gangotri temple area. We did not hike to Gaumukh. We just visited the temple area.
The common perception that the river Ganga is born at Gangotri is actually not quite right. It is the river Bagirathi that is born near Gangotri. Then, there are rivers Mandakini and Alaknanda born near Kedarnath and Badrinath respectively. Mandakini and Alaknanda flow south merge together at Rudra Prayag, and flow as one river with the name Alaknanda. Then, Bagirathi and the Alaknanda from Rudra Prayag merge at Dev Prayag further down south to become Ganga. Ganga has such a sacred stature presumably because it is formed out of merging of many Himalayan rivers.
The temple at Gangotri itself was hardly impressive to me for its size, sculpture or organization of pilgrim/tourist logistics. In the temple, the river Ganga is personified as a powerful female deity.
While the temple did not really wow me, the drive to Gangotri from Uttarkashi did give some spectacular views. The winding ghat road ran by the Bhagirathi river for its entire stretch. The road was single lane, but it had two-way traffic. There were steep drops, which did seem scary at places. Traffic also would standstill for couple of hours around certain narrow turns with the police having to rush to aid in clearing the traffic.