Simple battery backup auto-switch circuit

Anyone want to help me with a simple circuit design that switches from PSU to battery power when the PSU power falls out? I could use a relay, but I'm afraid it wouldn't be quick enough to keep the equipment up and running without trouble. I'm going to use it for keeping a switch and a router continously running, even in case of a power outage. They are running at 9 and 12 volts respectively, so I assume it will be easiest to make two separate circuits for the two.
What would be even better is a circuit that switches to an other power supply/battery pack when voltage drops below a certain point, then I could use it for switching between different battery packs as well. Would that be complicated?

My opinion: build a new power supply with 13.8V output, and connect a maintanence-free battery (accu) to output. (this type of battery designed for 13.8V continous charge mode) Now You have an UPS with 13.8V out.Apply an LDO for 12V device, and a normal (dissipative)regulator for 9V device.
Without current consumptios of Your devices can't rcommend exact types.

Thanks, but wouldn't that either continously draw power from the batteries even when the PSU is running, or apply DC to the batteries? :?
I guess your solution would make worthy power disappear in heat through the regulator, so I'm rather using separate battery packs for the two voltages.
Both of the devices are rated 1A at their respective voltage.