I have unclogged a lot of toilets and I only remember one time that I had to pull the toilet up to dislodge a uniquely shaped toy that was caught solidly on the back side of the trap. You can use a small piece of mirror to look up into the trap if you suspect something is caught. Sometimes the toilet will drain only to clog again if something is caught in it.
Removing a toilet to solve a toilet clog problem is a last resort. First try 'Plunging a Toilet', then try 'Using a Toilet Auger' before you consider removing the toilet. One other consideration is your main sewer line. At toilet that won't unclog may be blocked in the sewer line. See the article 'Troubleshooting Drain and Sewer Line Problems' to make sure that your problem is not more serious.
Removing a toilet will involve several steps and is not considered to be an easy job. You are also going to need to replace a some parts when you have the toilet removed. For instructions on removing a toilet, see the article 'How To Remove a Toilet'.
If for you see water in the pipe that is in the floor, it is not your toilet at all. Your sewer line is clogged and will need to be snaked out.