The original Bohr model of the atom was not much further advanced than the "plum pudding" model - that is, an atom was seen as a solid sphere dotted with stationary electrons. In the Bohr model, the atom is considered to be a nucleus of positive charge, with electrons as point charges orbiting the nucleus in a manner similar to the orbit of the moon around the Earth, or planets around a star. This model was based on considerations of the Hydrogen atom, and whilst it made some reasonably accurate predictions for Hydrogen, it fell apart for more complicated atoms.
Eventually, it was seen that it was more accurate to describe electrons orbiting a nucleus as waves delocalised over space, rather than point particles. From solutions to the Schrödinger equation, it can be seen that the regions electrons inhabit - or "orbitals" - are not all simply a delocalisation of charge density over all space, falling off as 1/r2 - such as with gravity. Instead, electrons in different quantum states have a radial distribution function which describes the shape of the region in which the electron might be found. It is notable that, other than at specific nodes, orbitals are not the extreme confines of the electron - like the edges of a helium balloon are the edges of the confined gas. Rather, the drawn shape of an orbital shows the boundary at which the probability of finding the electron is at some arbitrary value. This could be, say, a 0.9 probability. There is still a finite probability of finding the electron outside of this region, but this probability drops off rapidly - to all intents and purposes, the electron's existence outside the drawn region of the orbital can be ignored.
To see simulations of the atomic orbitals, please click here.