When the toilet backs up you would normally reach for your plunger. Sometimes you just don't have a plunger available. You can use several different things as a substitute for your plunger that will get you the same result--a clog free toilet that is running smoothly--at least until you can procure an actual plunger.
Dish soap is a good and fast way to unclog your toilet. This will only work if your toilet is not overflowing. If it is not, then bring a pot of water of water to a boil and squirt a cup of dish soap in the toilet. Carefully and slowly pour the boiling water into the toilet bowl. The soap will act as a lubricant to loosen the clog and the hot water will flush it away.
You can use a soda bottle full of hot water to unclog a backed-up toilet. This is a more hands-on way to unclog it and not for the faint of heart. Using a bucket and cup, remove most of the excess water from the toilet. Find an empty 20 oz. soda bottle. Turn the water in the sink on as hot as you can and fill the bottle with the hot water. Place your fingers over the opening of the bottle and tip it down into the opening of the toilet bowl drain. Remove your fingers and squeeze the bottle until the bottle is empty. You may need to repeat this several times to clear the toilet. The force of the hot water simulates the pressure differential of a toilet plunger.
You can use a simple wire hanger to unclog your toilet. Remove some of the excess water from your toilet. Find a wire hanger and unravel the ends so you have one long piece of metal, then bend your hanger in half and slide both ends down into the toilet bowl, being careful not to scratch your toilet. Once the hanger is inserted move it around in back and forth and circular motions to loosen the clog. Once you think you have loosened the clog enough, flush your toilet. If the obstruction is not completely gone you may need to try again.
You can also make a plunger of sorts using a 10-inch square of thin rubber. Put the rubber over the mouth of the bowl drain, and using a gloved hand, apply gentle in-and-out pressure on the rubber. This will create pressure differentials that will help loosen stuck material.
You may have to try each of these ways more than once to effectively remove the clog from your toilet. When dealing with a clogged toilet be sure to wear eye protection and rubber gloves to avoid injury or infection.