A toilet won't flush fully if the rubber flapper in the toilet tank closes too quickly. Make sure the flapper is hinged properly on the overflow tube. Then look to see if the flapper has an adjustment--if it does, make sure it's set up properly.
Check the chain's length--pull it tight, then back it off one link and re-attach it.
Flappers, available in hardware stores and home improvement centers, are very inexpensive to buy. If the above remedies don't work, replace the flapper with a new one.