A nebula is a giant cloud of dust and gas in space. Some nebulae (more than one nebula) come from the gas and dust thrown out by the explosion of a dying star, such as a supernova. Other nebulae are regions where new stars are beginning to form

Are planetary nebulae the result of supernovae?

No, they are two entirely different things. A planetary nebula is born when a low mass star dies (low mass means less than about 8 times the mass of the Sun), while the supernova is the death of a massive star.
In low mass stars, the outer envelope of the star is ejected out while the core of the star becomes a "white dwarf". The ejected envelope expands away from the central star and creates the nebula that we see. So, if you look at the pictures of classic planetary nebulae like the Ring nebula, you will be a ring of material with a star in the center. The ring of material was once part of the star in the center, but now has been ejected. The central star is now a white dwarf.

In high mass stars, the story is entirely different. Nuclear fusion in the core produces elements all the way to iron. Once the mass of the iron core exceeds a certain threshold, it collapses causing shock waves to propagate outwards. The end result is the titanic explosion called the supernova, leaving behind a neutron star or a black hole. The exploded remains of the star form a "supernova remnant". Examples are the Crab nebula (M1 in Taurus) and the Veil nebula in Cygnus. Hence, a supernova produces a supernova remnant and not a planetary nebula.

Orion Nebula

Hubble image of the Orion Nebula shows dense pillars of gas and dust that may be the homes of fledgling stars, and hot, young, massive stars that have emerged from their cocoons and are shaping the nebula with powerful ultraviolet light.

Cosmic 'DNA': Double Helix

his is a color enhanced version of the infrared signal to make the Double Helix Nebula's features easier to see. The spots are mostly red giants and red supergiants. Many other stars are present, but are too dim to appear.

Two-Star Collision Yields Three-Ring Nebula

Supernova 1987A occurred in the Large Magellanic Cloud, a galaxy only 160,000 light years from Earth. The outburst was visible to the naked eye, and is the brightest known supernova in almost 400 years.

Sun's Baby Twin

Located in the Eagle Nebula, E42 is thought to be a very early embryo of a star much like Earth's Sun.

Near-Perfect Symmetry - Cosmic Square

An image of the Red Square nebula surrounding the hot star MWC 922. The picture was taken with infrared adaptive optics imaging at Palomar and Keck Observatories.

Cosmic Hand Reaches for the Light

Red represents low-energy X-rays, the medium range is green, and the most energetic ones are colored blue. The blue hand-like structure was created by energy emanating from the nebula around they dying star PSR B1509-58. The red areas are from a neighboring gas cloud called RCW 89.