Put simply, Quasars are the extremely active centres of some, not all galaxies. Quasars are just one of a number of different Active Galactic Nuclei, the others include Blazars and Radio Galaxies. Quasars full name is QUASi-StAllar Radio object. The centre of the milky way is reputed to be just a supermassive black hole and not a quasar. Quasars are large supermassive black holes at the centre of galaxy that is so active, that it tends to outshine the Stars in the galaxy. Quasars tend to have accretion discs surrounding them like Saturn but on a grander scale. Quasars tend to be at the centres of galaxies far far away. The nearest Quasar is 3C273, it is 2.5 Billion light years away in the constellation of Virgo. Ref: N.A.S.A. . Some Quasars that are studied have formed not long after the Big Bang. The picture below is a N.A.S.A. artist's impression of what a Quasar looks like.
To look at a Quasar from Earth, they would appear as red stars. When I mean look, I mean through a telescope, they are too distant and from here, too small to be seen with a naked eye. The redness is because they are so far away and also because they appear in the redshift of the spectrum, that they are moving away from us. Quasars are similar to pulsars (Pulsating Stars) in that they have streams of xrays shooting from their middles. They don't rotate as often if ever. A quasar is approximately about 1 Kiloparsec in width which is 1000 parsecs which is 1 parsec = 3.26 Light years. Ref: Starchild
As Quasars are so far away, their light has been travelling for billions of years. Quasar ULAS J1120+0641, which has a size of over 2 billion times the mass of the Sun. It is believed to have formed only about several hundred million years after the Big Bang.Ref: Space
The difference between Quasar, Radio Galaxy and a Blazar is the angle of the stream. If the stream is straight up, it is a radio galaxy and we're not in the firing line. If the stream is angled slightly towards us, then its a Quasar and if the stream is angle directly towards us, its a Blazar. In the beginning when they were first discovered, they were thought to be different objects in space but as we learned more, we discovered they were the same. Below is a video talking about these types of objects.
The Milky Way is not a Quasar, Blazar or Radio Galaxy, however that doesn't mean to say that it has never been and never will be a radio galaxy. All the quasar etc that we see are at the end of the observable universe which indicates that they are younger than nearer galaxies because their light has had to travel a further distance. The further away an object is, the younger stage of life we see it. When the Andromeda Galaxy eventually collides with our galaxy, there is nothing to say, the resulting supergalaxy won't become an Active Galactic Nuclei galaxy. As the collision won't be around for a few billion years, we won't be around to see it.
The Big Bang Theory, the actual theory, not the television series is based on Hubble's Law. Hubble's Law was developed by Edwin Hubble in the early part of the twentieth century. He stated that when a galaxies light is spectrograph, split into colours of the rainbow, if the light is more heavily red, the item is moving away from the observer. If the light is more blue, the object is moving closer to the observer. However when observing Quasar light, the items are more red than they should be. Using Hubbles Law, they are calculated at being at the edge of the known universe and that they are moving at the speed of light.
As of August 2015, the nearest Quasar to Earth is Markarian 231 according to NASA Hubble space observatory. It is estimated to be about 600 million light years away in the constellation of Ursa Major in the northern hemisphere. It is believed that the galaxy has two black holes that are orbiting around one another. The light that we are seeing now is how it was 600 million years ago, that is even before the dinosaurs roamed the earth over 250 million years ago. The central black hole is believed to be 150 million times the mass of the Sun and weigh over 4 million times the mass of the Sun. Even though the quasar is so far away and old, it is believed that the two black holes orbit round each other 1.2 years. The reason why they are believed to be in orbit round one another is the result of two galaxies colliding. The two black holes will have already collided by now but we would not see the effect of the collision here on Earth for another few hundred thousand years. ref:N.A.S.A.