Energy of an electron (why negative?)
Let me explain you in the simplest manner. 

For a two object system,

If they repel each other when you move them towards each other, that means it takes work to do this, and the potential energy is positive.

If they attract each other when you move them towards each other, then potential energy of the two object system goes on decreasing.

Potential energy of a two charged object system at a particular orientation is defined as work done by external agent to bring the second body from infinity to desired point in presence of the field created by the first body. Now think the first body to be nucleus and the second body to be electron.  Now to find the potential energy, according to potential energy defination I just said above, we need to being the electron from infinity to any desired location, lets say to a distance of "x" from the nucleus. As we have to bring electron from infinite that means initially electron is infinitely far away from nucleus and hence at zero potential.

Potential Energy of electron:

When we are bring the electron towards the nucleus, offcourse they are going to attract each other and as said earlier if two objects attract each other, then potential energy of the system goes on decreasing. So as we move electron towards the nucleus, energy goes on decreasing. Note that initially as electrons and nucleus were at infinite distance from one another, hence initially at zero potential and any further decrease in potential energy implies negative potential energy. 
So potential energy of the electron is negative inside any atom. 

Kinetic Energy of electron:

The electron also has kinetic energy. Kinetic energy can never be negative (because in K.E expression we have squared of velocity which is always positive) and it is smaller in magnitude than the potential energy. 

Total Energy of electron:

So the total energy i.e kinetic energy  + potential energy is negative.