Do an experiment. Fill a bucket with water and slowly pour it into the bowl. The water level should only slightly increase, and when you stop, it will return to where it was when you began. If that works, dump a bunch of water in and see how well that works. While not exactly the same, it should flush. Modern toilets tend to use a siphon jet (the hole at the front of the toilet) to accelerate the water so it can start to siphon. Old toilets, just relied on filling the bowl with lots of water, and eventually, it would have enough pressure to start the flush. DUmping a lot of water into the toilet bowl emulates an older toilet since there's nothing going through the siphon jet path.
If your siphon jet's path is plugged, there's not enough water going through to cause it to flush.
It could be lots of things, but you may have a partially blocked drain line. This could be in the toilet, at the flange (excess wax, or improperly installed could be blocking part of the outlet), or down the drain line. A toilet won't flush if the drain line is full, or it can't move the water out properly.
Make sure the hose from the fill valve is properly positioned and the bowl is getting properly filled prior to the flush.