If I have two routers, how do computers choose which to use?

We have two ISPs at home, and I've connected a different router to each and both routers into our LAN, one large etheet switch that everything connects to. One router has the IP 192.168.1.1 and connects through Time Waer Cable, and the other has 10.0.1.1 and connects through Verizon. Both are DHCP servers as well.

For whatever reason, anything with DHCP enabled seems to take a DHCP lease from 192.168.1.1, thus accepting 192.168.1.1 as the router and connecting to the Inteet through TWC. I don't know if that's because that router just always "wins" a race condition or if there's some actual priority. How do computers choose which router to take the DHCP lease from?

I don't have enterprise-level hardware. The router on TWC is an Apple AirPort Extreme, and the one on Verizon is the Actiontec they gave us. I've been looking into "DHCP failover," but it always involves a Linux or Windows server managing DHCP. And when I Google for solutions involving multiple routers, I get people who think that router = wireless access point.

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