What is wrong with my central heating?
This may be as simple as the "start" or "run" capacitor going south (because the motor starts but dies afterwards, I suspect that the run capacitor is blown).

If you are comfortable digging around in the control box for the furnace, then you should be able to find 1 or 2 large-ish capacitors.  If you have a multimeter, you can test them against the specs that are listed on the casing.  They should be within 10% of the nominal value.  If you don't have a meter, then you can still give them a physical inspection.  Occasionally, they will bulge or leak or give some other indication that they have failed.  

If you find that either of them has failed, then you should be able to find a capacitor with similar specs online (or at a local HVAC supply).  It should not run you more than about $15.

Here is a link with additional info and a short video explaining more about the capacitors.