Will your phone work with the power out?
Old fashioned line-powered phones don't need an external power source. They get their operating power right from the phone line.
However, most of us have cordless phones with a base station and one or more handsets. Many of us use VOIP phones, such as Vonage, which use our Internet connection. No line power there.
The easiest way to ensure you have a working telephone during a power outage, if your phones are not line powered but you still have regular phone service, is to keep a line powered phone around. You can pick them up cheap at Lowe's or Home Despot for around $10 each. If you have VOIP service which depends on an Internet connection*, then you need a battery backup, AKA uninterruptible power supply.
You can buy UPSes meant for computers and networking gear at places like Staples, Office Depot, Microcenter, or CompUSA. Well known, reputable brands include Belkin, American Power Conversion (APC), and Tripplite. Prices will range from a low of around $30 to enough to melt your Master Card.
UPS capacity is measured in Volt Amps (VA). The more the VA, the longer you'll have power available. How long it'll last depends on what you have connected to the UPS. E.g., if you keep surfing the web on battery backup it's going to kill your battery sooner than if you have only your Vonage router and cable modem connected.
Note that some equipment will continue to draw power even if the power switch is off. E.g., most recent PCs have a power switch on the front of the case and a second one on the power supply in back. If you leave the PC connected to the UPS, it'll continue to draw a little bit of power unless you unplug it or turn off the switch on the power supply in the back of the case. I also recommend against plugging printers and scanners into a UPS because of the potential for draining the power, especially if the printer/scanner is on and the power goes offline when you're not home. Plug printers and scanners into surge protectors, which are plugged into mains power.
If a power outage lasts more than a day or two it's likely that the UPS will run out of juice. But if most power outages you experience are short term, a UPS is a worthwhile investment to keep your primary means of communication open.
* ISPs have their equipment on massive battery backups and/or backup generators, so they often stay online during power outages. ASSuming that the cables haven't been knocked down, you can usually get online if you have backup power.