If an electron is part of an atom, how can we separate the electron from the atom?. If we consider the electron part of an atom what happens after removing the electron,?. Whether what remains behind the separation of the electrons can still be referred to as an atom
A neutral atom with some its electrons removed (or extra ones added) is called an ionized atom, or just an ion.
As Chris said, when one removes an electron from the atom it is bound to, one creates an ion (in that case one creates a positively charged ion since the electron has a negative charge). Ions are still atoms but with a different electronic configuration. Ions are of great interest in science since they are part of plenty of phenomena in Physics, Electronics, Chemistry, Biology, Medicine, etc.
There are many ways to produce positively charged ions (that is to remove at least one electron to an atom). Basically, the idea is to provide a sufficient amount of energy to the electron in order to "free" it from the atom. In simple words, the electron has a certain amount of energy in being linked to the nucleus but if one provides a greater amount of energy then the electron can escape.
This phenomenon is called "ionization", you can have a look on the internet for deeper explanation.
Well, basically electron is not actually a part of an atom. It is in a bound state. There are various potential wells around the atom which contain electrons of different energy. If an electron is having energy = E_1 then the electron is in the 1st potential well. If it has energy = E_2 [E_2 > E_1] then it is in 2nd potential well. (These energy levels are quantized, i.e. You can write E_2 = 2*E_1)The no. Of potential wells depends on the amount of charge the nucleus has. If the electron gains energy E_n suppose such that this energy is greater than the potential of the last potential well then the electron escapes the atom quickly!
Basically this is the behavior of quantum mechanics. Vector model of atom. This is the reason why electrons do not fall into the nucleus as their energy is quantized due to the fact that they are in quantized potential barriers!!