Getting 12 V, 110 mA supply
The fact that the power supply is rated \$12\ V\$ at \$1\ A\$ only means that the power supply can output up to \$1\ A\$. The actual current output will be based on the requirements of the specific circuit.
If I put to that supply a circuit that draws \$110\ mA\$, as in your case, that supply will output \$110\ mA\$ at \$12\ V\$, which is a little over a watt of power. If I connect a circuit that requires more than \$1\ A\$ to function, that specific supply won't be able to give enough current to the circuit since the maximum current the supply can provide is \$1\ A\$.
So if I want to power a circuit that requires a \$12\ V\$ rail at \$2\ A\$, for example, I'll need a supply which can output more than \$2\ A\$.
In your specific case a \$1\ A\$ supply for a \$110\ mA\$ circuit is well over enough. You could even do it with a much smaller \$0.5\ A\$ supply if all you need is \$110\ mA\$.