In physics sometimes we find energy that is negative. What does the negative sign indicate? Sometimes we see energy that is negative, for example, the energy of an electron in orbit. We know energy is something that can do something. In this view does negative energy mean something opposite someway?
 A: Absolute energies have no observable effect, it is only changes in energy that have physical meaning. How to interpret a negative energy depends on the context.
In the example you give of an electron bound to an atom, the energy is measured with respect to the electron being free from the atom and infinitely far away. So the binding energy being negative means that the bound state corresponds to a lower energy than the isolated state, and thus you would need to supply the bound state with energy in order to isolate the electron.
A: Take the simplest atom, the H atom. A hydrogen atom consists in a proton and an electron. When a proton and an electron bind to form a hydrogen atom, there is a release of energy. So in order to break the atom into its constituent parts, one has to provide an amount of energy. The amount of energy required to break the atom into its constituent parts is equal to the amount of energy released when the atom has formed. This is considered negative because you have to give energy to the system in order to break the bound. When it reaches zero, the particles become free.
A: Negative sign simply means that the system is releasing energy. For instance, take the case of gravitational force. The potential energy function of gravity is,
$$U_g(r)=-G\frac{m_1m_2}{r}; U_g(\infty)=0$$
Gravitational potential energy is described as the work done in bringing a body of mass $m_2$ towards $m_1$ from $\infty$ to a certain separation $r$. Since, the two bodies pull each other, no external work is done on the system. Instead we obtain work from it. This explains the negative sign.
