Q:

Does Simultaneous dual band routers mean they have dual radios for each band? 

We are trying to construct a wireless mesh network.  We currently have single band routers that operate on 2.4GHz.  We have also succeeded in meshing the two routers. 
We have chosen B.A.T.M.A.N Advanced layer 2 mesh protocol.  Therefore for mesh devices we do not need IP address to communicate.  Currently the router is flashed with OpenWRT and operates on two modes simultaneously. 

Adhoc mode (for mesh) forms the backbone network. 
AP mode (for allowing non-mesh devices to connect via layer 3) for client network. 

We know this setup with only single band router with single radio will be unusable over the long run since the same radio have to operate on both adhoc and AP mode at the same time causing interrupts. 
We were looking out for dual radio routers and found a lot of dual band routers, so that 2.4GHz can be used for client networking in AP mode and 5GHz band can be used for mesh backbone in Adhoc mode.  Therefore two different channels. 
What we are confused at the moment is, does the simultaneous dual band routers mean they have dual radio (or) they manage using single radio? 

A:

Yes, dual band routers mean they have dual radio and transmit on a 5 GHz band and a 2.4 GHz band.  They operate on two different frequencies and contain at least two radios with accompanying antennas tuned individually to two different wireless frequency bands.  One radio signal connects to the 2.4 GHz band, and the other connects to the 5 GHz band.  The 2.4 GHz access point uses the 802.11b, 802.11g, and 802.11n frequency bands, while the 5 GHz access point uses the 802.11a frequency band.  The primary purpose of dual band routers is to speed up wireless connections, and they do that by supplying separate network bandwidth for different devices.