Here are the types of objects we can observe in the night sky. I won't go into any more detail on subcategories at this time.
Deep space
Stars
The beginning of everything, the stars. They are spheres of gas condensing which by the nuclear reactions of the elements that compose them emit their own brilliance. There are various types: dwarfs, giants, reds, supernovas, etc.
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Galaxies
They contain an enormous amount of stars and stardust which are united by a great gravitational force that generally comes from the nucleus. To give you an idea of how many stars they have, a "dwarf" galaxy, or one of the smaller ones, has a billion stars. The center of these objects is a very massive object, presumably a black hole. Galaxies are classified into spirals, ellipticals, and irregulars.
Nebulae
They are clouds of gas and stellar dust that can be the result of a star's explosion or simply the accumulation of such elements in space.
Open clusters
It's a group of stars not as massive as galaxies but sharing the same gravitational space. The number of stars it contains is infinitely less than that of a galaxy. In fact, a galaxy can contain several such clusters.
Globular clusters
Exactly the same as open clusters but the number of stars is much higher. Sometimes they are so massive, like the Omega Centauri, that they are totally spectacular.
Solar system
Planets
Planet means "wanderer." If you see something like an unexpected star in the sky that you don't remember is part of the constellation and its trajectory differs from that of the stars, you have found a planet. All of them are in our solar system. With a telescope we can see Mars, Jupiter, Saturn, Venus and Mercury.
Comets
They are loose rocks or pieces of ice that are attracted by the gravity of our Sun to our Solar System. Many of them are believed to come from an area called the Kuiper Belt which is an area on the outskirts of our Solar System.
Ison Comet
Our Sun
With special telescopes, we can observe our Sun, including its spots and flares. 
Luna
This is the easiest object to observe, not so much to photograph. It reflects the light of the Sun so much that filters are needed to take pictures of it. 
