Connecting 12v 1A device
You can connect it to your device rated at 12V 1A. If given the proper voltage (12V) the device will not sink more than 1A even if the battery is capable of giving more current

In the same manner, your home AC network is connected to a variety of devices, TV, fridge, microwave and phone chargers, each one sinks a very different current and each one is connected to the same AC mains capable of giving much more current than each of them can sink.