Runners have got to be some of the whiniest people I've ever met. When it's cold, I swear half my Facebook friends have some status up about how miserable it is to finish a run with icicles on their eyelashes or how painful it is to defrost in the shower after a run. When it finally gets above freezing, the statuses change to how much everyone hates cold rain. Then it gets hot out, and then everyone starts complaining about the heat and humidity. Don't get me wrong, I love perfect weather as much as anyone, and I've been known to whine mightily when it's gross out. However, you don't have to run for long before you realize that the gnarly days far outnumber the beautiful days, and the chances that your goal race falls on a day with absolutely perfect weather is slim to none (and if you only train on beautiful days, you're not going to be very good). Luckily, some bad weather is "less bad" than others (i.e. -5 °F and windy is better than 35 °F and raining, and both are better than 60 °F and hailing), and most of it can be combated with proper clothing, correct hydration, and other preparation. Like any other less than ideal weather, hot weather running brings with it certain challenges.
Running in the heat can be particularly challenging because exercise generates internal heat, in addition to the heat coming from the external environment. Even under cool conditions, body temperature can rise with exercise. You can imagine how much more quickly this happens in hot weather, and at a certain point (approximately 102 °F), performance declines. As the body heats up, fluid is sent to the skin, and through evaporation of sweat, the body is able to cool itself. However, this leads to dehydration and means that there is less blood available to provide muscles with oxygen, which is not good for performance either. In addition to having a negative effect on performance, heat can also be dangerous to run in, with hazards such as heat cramps, heat exhaustion, heatstroke, and other heat related injuries.
Luckily, we do acclimatize to heat, a process that takes between 7-14 days, though improvements may occur for up to 30 days. Ideally, workouts should be done in a range of intensities, temperatures, and humidities in order to fully acclimatize. A heat acclimatized runner will see changes in heart rate, increased resistance to raised body temperature, more efficient cooling through sweating, and lower salt concentration in sweat. Heat acclimatization is retained fully for about a week after returning to cool temperatures, and is thereafter lost gradually throughout a period of about 28 days.
The best tip I can give you about running in hot weather is to run early before it gets hot, or to run late after it cools down. Anyone doing doubles has already figured this out, but those of you who are used to running over your lunch hour may want to rearrange your schedules a bit. In general, mornings are cooler but more humid, and evenings are warmer but less humid (humidity is important because sweat evaporates more rapidly in a less humid environment, allowing more efficient cooling). Ideally, you can train at the time of day your goal race will be run, though unfortunately, not everyone's schedule affords that kind of flexibility.
The other important part of the heat puzzle is staying hydrated. A drop in fluid level means that the body is less efficient at cooling itself. Additionally, a 3-5% decrease in body weight due to fluid loss is enough to have a detrimental effect on performance. It is possible to become overheated without being dehydrated, and to become dehydrated without becoming overheated. Since different people sweat at different rates and have different tolerances for how much liquid they can take at one time, it is important to be cognizant of your own personal fluid needs (though in general, people replace less fluid than they lose during exercise). Another thing to consider is that during hard efforts, it may be beneficial to use a sports drink instead of water, in order to prevent hyponatremia (low sodium levels in the blood). One last important point is that people on a low carbohydrate diet are more likely to become dangerously dehydrated since water is stored with carbohydrates (I would very much discourage a low-carb diet when trying to train for a number of other reasons too, but this one is a biggie in the summer).
Dehydration can be combated by making sure the athlete is properly hydrated before exercise, as well as taking in fluid during exercise. Before exercise, the athlete should drink enough that their urine is very lightly colored, but not so much that they're overhydrated and peeing every half hour, which will cause a loss in sodium and potassium. Hydrating during exercise can be a little tougher in that the athlete needs to figure out where to get their liquid from. If you're lucky, you have a coach (or friend or significant other) who will follow you on a bike or meet you at certain points in a car. For everyone else, storing water bottles along your route or doing short loops that run past your house is probably the easiest way. Ultrarunners may want to practice running with a handheld (or FuelBelt or Camelbak).
Clothing choice is also important. Every so often, it'll be ridiculously hot out, and I'll see someone huffing along in a sweatsuit. You do not get a better workout by wearing clothes that make you sweat more. For the most part, light-colored, loose clothing made of porous materials that do not stick to the skin is best, and often, the less clothes the better (within reason, of course). Some people like wearing a light-colored hat, while others find that it makes their head hotter. On very hot days, I'll wear a hat and periodically douse it with cold water, since I'm either doing small loops or running past a water source on those days. Of course, there are always some exceptions to this rule, such as when training for extremely hot races (we've all heard stories about the elite runner who trained for a hot race wearing a sauna suit), and when running in extremely hot environments (check out the clothes that Badwater competitors wear), but that's really the exception, not the rule.
One other thing to consider doing in the summer is sponging. Hot weather causes blood to pool in the veins of the arms and legs as the veins dilate. Lowering the skin temperature through sponging will constrict these veins and send the blood back to the heart, increasing the volume of blood filling the heart and aiding the heart in maintaining a high stroke volume and cardiac output at a lower heart rate.
And if all else fails, there's always the treadmill for those really hot days.