I will be running a dynamic web site and if the server ever is to stop responding, I'd like to failover to a static website that displays a "We are down for maintenance" page. I have been reading and I found that switching the DNS dynamically may be an option, but how quick will that change take place? And will everyone see the change immediately? Are there any better ways to failover to another server?
DNS has a TTL (time to live) and gets cached until the TTL expires. So a DNS cutover does not happen immediately. Everyone with a cached DNS lookup of your site still uses the old value. You could set an insanely short TTL but this is crappy for performance. DNS is almost certainly not the right way to accomplish what you are doing.
A load balancer can do this kind of immediate switchover. All traffic always hits the load balancer first which under normal circumstances proxies requests along to your main web server(s). In the event of web server crash, you can just have the load balancer direct all web traffic to your failover web server.
pound, perlbal or other software load-balancer could do that, I believe, yes
perhaps even Apache rewrite rules could allow this? I'm not sure if there's a way to branch when the dynamic server is not available, though. Customize Apache 404 response to your liking?