Suppose you suspected a certain object in the sky to be a quasar. What observations would you need to confirm your hypothesis?
You would look for the general characteristics of quasars (which overall are unique Active Galactic Nucleoluses), such as the facts that they are;
They are the brightest objects in the universe
The spectrum of most quasars is noticeably red shifted (between z=0.056 to z=7.085,
They are commonly very old (we can deduce the approximate age of the universe from quasars)
The spectrum of adjacent stars is abnormally shifted because they are orbiting a supper massive black hole
10% of the time the Quasar will have lobes or jets of material streaming from the direction of it's poles (as shown below):
A quasar is typically a very distant galaxy emitting massive amounts of radiation.
Most galaxies have a supermassive black hole at their centres. In the case of young galaxies, or galaxies which have collided with others, there is a lot of dust and gas near the black hole.
The dust and gas starts spiralling into the supermassive black hole forming an accretion disk. As more material falls into the accretion disk it gets superheated by friction and gravitational effects. This causes massive amounts of radiation to be emitted. This is a quasar.
Quasars are typically very distant objects. They have characteristic very high red shifts as a result. They are very young galaxies from the early universe. Their distance means that their light has taken billions of years to reach us. All distant quasars will long ago have run out of material around the super massive black hole and ceased to be quasars.
Quasars emit huge amounts of energy. Typically their output is about 100 times that of our galaxy. This energy output is confirmation of a quasar.
Quasars tend to be quite short lived. Though if a cloud of material gets close to a central supermassive black hole it could reignite as a quasar. Our galaxy has probably been through a quasar stage.