I noticed most Wi-Fi routers come with an 'Automatic' channel selection mode, instead of requiring the administrator to select a channel between 1 and 13.
Is there a standard way for the routers to determine which channel to use?
No, there's no standard. The IEEE 802.11 spec doesn't address automatic channel selection in the 2.4GHz band. The Wi-Fi Alliance's Wi-Fi logo certification testing programs don't address it either.
Different vendors have different proprietary algorithms for how their APs automatically select channels. Some take into account other Wi-Fi networks and pick the least congested channel, some limit the channel they will pick to 1, 6, or 11, some take into account the noise level on each channel (so they avoid channels with non-Wi-Fi interferers). Some weight the channels in the middle of the band more heavily than band-edge channels, because transmit power is often more limited on band-edge channels to avoid out-of-band emissions.
Some only do automatic channel selection at boot time, while others will change their channel in the middle of normal operation if the channel they were on becomes cluttered and is no longer the best choice. Some enterprise-class gear has ways for APs to coordinate with each other or via a central controller to pick channels that don't interfere with other APs on the same roaming network.