The town center in Age of Mythology, much like AoE and AoK, performs several duties. It trains villagers, researches upgrades, and advances ages. In AoM, it's also where the Greeks and Egyptians train heroes, where the Norse train ulfsarks, and where the Atlanteans train oracles.
Town centers also supply 20 population, and since many units take more than one population slot, population is very important. You can only build a maximum of 10 houses, which each supply 10 population. Atlanteans build manors, which are much like houses, but support 20 population and have a lower build maximum of 5. They also cost more.
Differences to AoE and AoKEdit
Unlike AoE, in which town centers don't attack, and AoK, in which they won't attack unless garrisoned, town centers in AoM will attack. They won't fire very many arrows unless garrisonned, but the attack and range is pretty good. Also unlike AoE and AoK, town centers can only be built on settlements, a premade base that cannot be destroyed.