Try holding the lever until the toilet flushes completely. If this doesn't work, then you need to adjust your lever to flapper connection so that pressing the lever pulls the flapper up further, allowing greater water flow.
If holding the lever down for awhile until the toilet flushes completely works then it is likely to do with a mismatch between your flush mechanism and your toilet's water requirements. Recent toilet designs in the US must comply with government standards for water use, and are now low flow. The new toilets are designed to work well with less water, typically 1.6 gallons per flush or less. This is under half the water old toilets required. If the toilet is older than 1994, then it likely is a style that requires 3.5 gallons per flush.
If you've replaced the top tank, or the flushing mechanism without replacing the old toilet bowl, you've run into the same problem many people have run into since the changes have been made. Your old bowl will not flush properly with the low amount of water your new flushing mechanism provides.
However, manufacturers have designed flushing mechanisms to allow the user to provide a greater flush volume if desired. You simply hold down the flush lever continuously until the toilet is completely flushed, using up the entire tank of water.
It's not as convenient as pressing and releasing immediately, but it should get the job done.
Another option is to purchase a 3.5 gallon per flush mechanism and replace yours with the larger volume flush mechanism.
Alternately, you can replace the toilet bowl with one designed to work with a low flush volume.
If you want to modify your existing flush mechanism to provide adequate water volume permanently, you can examine the mechanism and you should be able to modify it so it empties the entire tank into the bowl with a single press and release. In most mechanisms the low flow is accomplished with a small hole in the flapper. When the flapper is raised by the lever, there is an air bubble trapped in it that keeps it up until the water level drops low enough. In low flow mechanisms there is also a small hole that releases the air in this bubble, so the flap drops before the water reaches the bottom of the tank. Block this hole, and the flap will stay up until the tank is emptied.
The combined overflow tube and flapper system you have should have a similar small hole that allows the flapper to drop before the tank is emptied, although it might not be as obvious as it is on a typical flapper.
Another method that is used is a separate float on the chain or flapper assembly that is higher up, which allows the flapper to close before the tank is fully empty. These can be adjusted downward to allow the tank to empty more fully, but they may not be able to be adjusted down far enough to fully empty the tank.
If you examine the mechanism while it's flushing, you might be able to determine what is allowing your flapper to close before the tank is fully empty.
Note that if you do make such a change to your system, you should not be in violation of federal standards, but local regulations may prohibit such modifications.
However, in all cases, holding the lever until the toilet flushes completely should work, unless the mechanism isn't adjusted properly in the first place.
The one time this happened to me it turned out that my 1.5-yr-old had flushed a quarter-turn piece of his hamster run into the toilet.
The snake went through it and we couldn't find any obstruction.
So we decided to replace the toilet.
As soon as it was off the floor we turned it over and there it was. A nice piece of blue curved plastic.
It was wedged in such a way that the snake made it pivot to go through, but water flow would catch in it and it would turn back to block the flow.
So the answer in our case was to remove the toilet and give it a thorough examination.
Best Answer
Sounds like the flapper is leaking and/or broken altogether. See instructions on replacing a flapper here: http://www.wikihow.com/Fix-a-Flapper-on-a-Toilet
See other possibly useful answers here: How can I fix a toilet that flushes incompletely? and here: Why do my toilets flush slowly and often incompletely?
EDIT: Um, after re-reading the question, I wonder if the toilet isn't simply clogged. Does the tank re-fill and water stop flowing into the tank after flushing (you can hear when water is flowing)? If the tank re-fills and water shuts off, it isn't the flapper. If the tank re-fills and water stops, then it sounds more like the toilet is simply clogged. Did you try a plunger? If that does work and the tank does refill properly, then you'll need the drain snaked I think.