What are some easy ways to stay cool during the day without using electricity?
I think the first thing that comes to mind is what you can do to decrease the temperature around the home in the summer. I am here in Jakarta, Indonesia at the moment and I have found that covering the patios with container gardens has a remarkable impact. Beyond this, well-placed trees, and the like, can help a lot. If you compare the house where I am living to my mother-in-law's place across the street, without air conditioning there is no comparison. Even in the middle of the day my house is quite a bit cooler.
When I am in Chelan,Washington the summer temperature is often hotter still, but can do ok without an air conditioner if needed (we have one smallish window-mount).
The second thing is to pay attention to air flows. A couple of open windows can be used to generate breezes. Air flow is absolutely important.
Cold slowers/baths are good. I would add that sometimes the simple act of putting water on one's head can help a lot. If you live near a body of water, swimming helps too.
The bad news is that if it is hot enough none of this is "easy." It takes effort, paying attention to detail, and the like.
Another approach is to leave windows open at night and close them in the morning for as long as you can, closing blinds etc. to reduce the amount of sunlight-driven heat in the home.
In the end these are all techniques. What you do to make it all work depends on a combination of them.
This is true as much in winter as in summer: keeping shutters closed. It will reduce thermal exchanges, and keep a cool temperature inside.
Windows are the main surface for heat exchange in an house/appartment.
Solar gains account for over half of a typical home's cooling requirements in the northern United States, so shade your windows--preferably from the outside, using awnings and plants. While curtains and window shades will help some, sunlight coming through the window will heat the shades. When the shades heat up, they will transfer some of that heat to the air in your home. Light-colored roofing materials will also make a difference.
If you can't take major steps like that, a wet bandana can make a big difference in how you feel.
I am in Thailand now, it is the hottest time of year, relentless 40C and sun every day. No air conditioning.
I survive by staying indoors during the hottest time of day, getting my clothes wet. Sitting close to the ground can help. I find ceiling fans unhelpful as they bring down the hotter air that has risen to the ceiling.
Your body adapts to heat after about 2 weeks so it is good to let yourself get as hot as you are comfortable with. This is why air conditioning is bad, people overuse it and are less able to cope with heat.
Use the middle of the day for relaxing, sleeping or other non physical activity.
The locals here play cards when it is hot.
I have gone for an entire summer without using electricity. The fastest way to cool down is to take a cold shower or to keep your bath tub filled with water and take a dip when it gets too hot.
Also it is sometimes difficult to sleep when it is both hot and humid. To solve that I had to step out of the cold shower without drying off, and falling asleep while wet before it got hot. I would sleep right through the night.
Around 4:00AM the humidity would drop until about two hours after sunrise. This allowed for cool blissful morning breakfasts.
To keep body cool on hottest summer days, I use wet tee-shirt so as to mimic body's natural cooling system e.g. transpiration. Vaporisation of water really helps to feel good even under full sun. For the sake of not using too much tap water I found empirically that wetting at neck and upper back is most water-efficient while achieving good results.
At night it is possible to get asleep while wearing such wet clothes too.
Keep in mind common cotton wet clothes let most UVs pass to skin. Use anti-UV wet clothes or UV-cream under wet clothes under the sun.
Of course it is necessary to wet clothes once every hour or so.
I agree with the above answers but also depending on your location the results can vary. If the air is hotter outside than inside, opening windows can hurt your efforts. If this is the case and your house is in direct sunlight covering the windows during the day can decrease the amount of absorption significantly. If your house has many floors it is crucial to pay attention to "heat rises". Directing the flow from a lower floor to the top open windows will push the heat out.
Cooking is also an often ignored heat source. Barbecue, use an outdoor oven etc can help. And paying attention to other electronics as heat sources is important. Turn off and unplug what you aren't using.
If your house has a flat roof, cover it with water. This will of course require relatively large amounts of water, so the total energy requirements (not just yours) may not be less.