When dealing with a clogged toilet, there are a wide variety of solutions out there. However, the most common solutions aren't always the most effective. Sometimes, you might need to buy a new tool or try a slightly more potent solution. Here are some alternative solutions that you can try if your first attempts failed:
If a plunger didn't work out and you are still looking for a direct solution, then a plumbing snake could be perfect. You probably won't have one lying around at home, so you will need to go out and buy one, but most home improvement stores should carry them.
To use a plumbing snake, you essentially just feed the flexible end down the toilet until you feel that you have reached the blockage. You are then going to twist the end of the auger, rotating as you go. This will essentially cause the tip to act as a drill, forcing its way through the blockage. Once it breaks through the blockage, you are pretty much in the clear.
After the blockage has been broken, it should drain very easily. If it does manage to reform further down the drain, you might need to call a professional.
While you can try baking soda and vinegar for a safe chemical reaction, not every blockage will be removed by such measures. You might need to resort to much more dangerous chemicals, including drain cleaners. Thankfully, these can be purchased at essentially any store that sells cleaning products, including most grocery stores.
Before you actually use a drain cleaner, you should check and make sure that your situation calls for it. Drain cleaners aren't very good at handling solid and hard obstructions, but they are very good at dealing with soft and organic targets. If you are dealing with a hard blockage, breaking it up with a plumbing snake might be more effective.
To actually administer the drain cleaner, simply follow the directions on the bottle. Make sure that you close the lid of the toilet if possible, since the fumes of the chemicals can be quite harmful. Also make sure that the fumes and the chemicals themselves are gone before you try to work on the toilet again. To make this possible, you might want to set up a fan and open some windows, in order to flush the toxic fumes from your building as quickly as possible. If you have a rough time clearing the clog on your own, contact a company that specializes in plumbing services.