Make a cold compress with ice as quickly as possible. Your burned skin needs to cool down right away - just make sure to put a towel in between your skin and the ice.
Source: Skin Cancer Foundation
Drink LOTS of liquids. The burns draw fluids to the skin's surface, leaving the rest of your body thirsty and dehydrated. So be sure to lots of water.
Source: Skin Cancer Foundation
Repeatedly apply non-petroleum, moisturizing lotion while you skin is damp. This will keep you skin moist for several days and prevent it from drying out. Avoid petroleum lotions because they trap heat and make the burn worse.
Source: Skin Cancer Foundation
You can also apply a lotion, spray, or ointment that has 1% hydrocortisone. It is a topical steroid that can soothe skin irritation, pain, and swelling.
Sources: Prevention and Summit Medical Group
Another burn remedy is aloe vera. Known as a "burn plant," it can help heal your skin after mild burns and sunburns. You can peel the leaves of the plant and apply the sap directly to your skin or use an aloe vera lotion. The oils from the plant will reduce itching and stinging, as well as lessen the chances of peeling.
Source: UAMS
Lotions and aloe vera are widely considered the best home treatments for a sunburn, but if you don't have any in the house, there are some home-remedies that can provide temporary relief. One home remedy you can try, for example, is to make a body wrap with gauze and milk. The coolness will ease the heat, and the milk will create a layer of protein to protect your skin from further discomfort.
Source: Everyday Roots
Raw potatoes (and potato juice) are also sometimes recommended as quick home remedies for a burn. Potatoes have starch-based compounds that could help provide some temporary relief for the pain and reduce the stinging.
Sources: Everyday Roots and Mercola
You can also soak tea bags in cold water and place them on your eyelids. The tannic acid might help decrease swelling and relieve pain.
Source: Prevention
Another popular folk remedy for burns is to sprinkle baking soda into lukewarm bathwater. The baking soda dissolved in water can create a "cooling sensation" on the skin by drawing some of the heat out of the burn. However, it is important to remember that this will only provide temporary relief and is unlikely to relieve the pain for any significant period of time.
Sources: Prevention, Botanical and ZocDoc.
One thing that will definitely help is to wear loose clothing while your burn heals. This will help you to avoid any more irritation.
And... if you have to, go see a doctor. This is necessary when you have severe blistering over a large portion of your body, or if you get a fever, chills, or feel woozy and confused.
Source: Skin Cancer Foundation