What is the difference between how a rocket achieves flight and how an aeroplane does?

assuming you mean a jet plane, the only difference is that the plane gets oxidizer from the atmosphere around it, while the rocket brings its own.

both jets and rockets work by igniting fuel mixed with oxidizer. an oxidizer is something containing oxygen, so that the fuel can burn, since nothing can burn without oxygen.

in a jet, this is often kerosene mixed with air. in a rocket, this is often liquid hydrogen mixed with liqiud oxygen.

the jet gets air from outside through air intakes, while the rocket gets the liquid oxygen from a tank it brings along.