"Twinkle twinkle, little star. Power equals I squared R"
So great, but what does this have to do with our 1/4W resistors that we all know and love? Every electrical component has a maximum power rating, which is how much power the manufacturer claims it can handle before melting, exploding, catching fire, releasing its magic smoke, etc. Going on YouTube, it's easy to find countless demonstrations of components dramatically exceeding their power ratings.
Example: As seen in this picture, we have two 10k 5% (brown-black-orange-gold) resistors. We want to use these in a circuit where we know that they'll have at most 10V across them. Which, if either, can we use without having to worry about them catching on fire?
In the last form shown, power = voltage squared divided by the resistance, so 10V * 10V / 10,000 ohms = 0.01W = 10mW. Seeing as how these are rated for 0.25W and 0.125W, respectively, we can use either and never have to worry about them catching fire.
On the other hand, if we later decided we needed to replace these 10k resistors with 1k resistors, running the calculations again show that P = 10V * 10V / 1,000 = 0.1W, which still technically is less than either, but 0.1W is getting pretty dang close to 0.125W, so that 1/8W resistor is going to start getting pretty dang hot and probably quite a bit less reliable.
These power ratings are nominal ratings. When the manufacturer says that their resistors can handle 1/8W, they mean it can handle 1/8W in some specific scenario, usually involving 25C ambient temperature and unrestricted air flow around it for cooling. Leave something in a car, and ambient will rapidly go up, and lint, dust, and sleek packaging inevitably interfere with air flow around components. Exactly how close you can drive components to their specified power ratings will always depend on your specific project, but I generally try and keep them at least 25% away from their upper limits. Every engineer will have their own preferences on the matter, so don't bother trying to reach a consensus.
If you go someplace online like Digikey and poke around in their resistor catalog, you'll probably be able to see that 1/2W resistors are generally more expensive than 1/4W resistors, since people generally use less of them, they're bigger, etc. It's actually not unusual for it to be cheaper to buy several 1/4W and solder them together to make something that can handle 1/2W. If you see a lot of 1/4W resistors which all seem to be really redundant in a circuit, this very well may be why. Something to consider as we continue to talk about higher power resistors.
Poking around in our electronics catalogs, you should notice that prices have now started to climb into the $0.25-$3 range. Compared to the 1-3 cents per 1/4W resistor, this starts to hurt, so you want to make sure you really need these high power resistors before you start throwing them in a project all willy-nilly.
Granted, that's not quite right... (Thanks Mats in the comments for reminding me) Resistivity isn't constant over temperature, which is part of why using the proper power rating of resistor is so important. If you unscrew a 100W light bulb and put an ohm meter to it, you'll get something more in the range of 10 ohms. Rather odd, until you consider that tungsten has a positive thermal coefficient of resistivity. As soon as you turn it on, the metal filament gets hot and the resistance very quickly rises to the 144 ohm level you'd expect. If you think about it, this is why you usually see light bulbs fail when you first turn them on.
Of course, even this behemoth mason-jar-resistor can be dwarfed by many power resistors, which can grow to the size of a roll of paper towels for really high power needs. In the past, I've worked with power resistors the size of your forearm in the 64VDC electrical system of diesel-electric locomotives.
Rest assured, you can always find a bigger resistor if you need one. It just might get really expensive...