Are there stars outside the White Dwarfs, Main Sequence, and Giant and Supergiant? if so, what are they called?
Brown-Dwarfs, Black-Dwarfs, Red-Dwarfs, Neutron Stars.
There are other types of Stars such as Brown Dwarfs, hypothetical Black Dwarfs, Red-dwarfs, Neutron Stars, Pulsars, Magnetars.
Brown Dwarfs are Sub-Categorized according to their temperature ranges. Stars are Normally categorized according to their temperatures with O being the hottest followed by B then A, F, G, K and finally M. Our Sun is a type G-2 Star. A slightly hotter Star would be a G-1, but the discovery of Brown Dwarfs took this list even further with type T, L and Y brown Dwarfs.
Brown Dwarfs are considered failed Stars i.e. they are far too big to be considered Planets but they are still not big enough to be considered Stars. Astronomers believed that an object the 75 times the mass of Jupiter would just lack the mass needed to fuse hydrogen into Helium. A Brown Dwarf will start the same way as a normal Star but instead of going on it will stop and begin to cool down. Gliese-229B is an example of a Brown Dwarf.
Black Dwarfs are hypothetical i.e no one has observed them yet as it is considered that when a White Dwarf cools down in about a 100 Billion years it will stop emitting light and turn black and since the Universe is still far too young compared to a 100 Billion years they are almost impossible to detect.
Red Dwarfs are the most common type of Stars in the Universe, they are small and have low temperatures which makes them shine in the redder portion of the visible Spectrum. Red Dwarfs are low mass Stars which means that they burn their fuel very much slowly then Stars more massive then them. It is believed that Red Dwarfs can live up to a trillion years and since the universe is still very young no Red Dwarfs have seen to die yet. The Nearest Star to the Sun, Proxima-Centauri is a red Dwarf at 4.25 Light Years with a Surface temperature of about 2768.85 Degrees Celsius.
Neutron Stars are the left over matter from the death of a massive Star, a star with a core with 1.4-2.8 times the mass of the Sun. When a very massive Stars ends its fuel after consuming hydrogen, helium, carbon, magnesium, neon , oxygen and a bunch of other heavier elements fusion reactions Stop and the inward acting gravity of the Star collapses it on it's core, the pressure inside the core increases and so does the temperature such that even electron degeneracy pressure couldn't even stop it from further collapsing, the Star collapses even further such that the pressure is enough to squeeze electrons and Protons together to from a Neutron and a Neutrino after a while the core is filled with Neutrons and the core becomes Stable to give you a Neutron Star.
Pulsars and Magnetars are typically Neutron Stars. we are gonna need a separate question to explain Neutron Stars in more detail. Pulsars are fast rotating Neutron Stars with beams of light coming out of them, rotating about 100 times per second.
In certain Neutron Stars the Magnetic field can be stronger than Normal these are called Magnetars.