DNS-based solutions are one solution to this problem, but they are subject to ISPs that ignore TTLs and cache your DNS responses for improperly large periods, thus torpedoing your failover.
You can arrange for an IP range to be properly highly-available over multiple links to independent ISPs, but it's painful, expensive, and requires an expert network admin. In essence, you get some PI IPv4 space and an ASN, plus a professional-grade router, hook up to ISPs that support BGP, and advertise your own addresses to the internet-at-large.
If any of the above confuses you, you should not attempt to do this; it is much easier and cheaper to work with your existing ISP to arrange redundantly-routed connections to your premises. Any large ISP will be doing BGP with their peers, so their connections to the internet are unlikely to fail completely (if they do, you should certainly get a new ISP). They will also have multiple local POPs, and should be able to arrange for the links to your premises to route through completely disparate infrastructures.
One notable advantage to taking this path is that the ISP will usually give you a big discount on the second link, since it only ever has to carry traffic when the first one's down.
