That is a really good question and the answer is not so simple.

First of all, electrons are attached to the nucleus of an atom by the electromagnetic force. The protons in the nucleus are positively charged and the electrons are negatively charged so there is an attractive interaction between them. To understand this better, youll need to study electricity and magnetism.

Secondly, the hard part to understand is how the atoms are arranged around the nucleus. A lot of pictures show electrons orbiting the nucleus of an atom the way planets orbit the sun. However, this picture is incorrect. A better way to think about atoms is to think of the electrons whizzing quickly all over around the nucleus so fast that it is almost like they are everywhere at once. However, this picture is still not totally correct. What is really happening is that for each electron there is a certain volume around the nucleus where there is a probability of finding that electron. We call this volume an orbital. You can kind of think of it as a swimming pool for the electron. If you go swimming in the dark (dont actually do this), we dont know exactly where you are in the water, but we do know you are somewhere in the pool. It is a little more complicated than this for the electron though because it does not have to move from one place to another by swimming like you do. It would be as if one moment you were in one part of the pool and the next moment you were somewhere else in the pool.

To keep things simple, lets think about hydrogen. Since hydrogen only has one electron, we only need to worry about keeping track of one electron. The electron can have certain amounts of energy, and the amount of energy it has determines the shape and size of its orbital. To find the orbitals for the electron, we need to solve the Schroedinger equation which is an important equation in quantum mechanics. The solution to the Schroedinger equation tells us what amount of energy the electron can have at each orbital, and what the orbitals look like. The amount of energy it has is called an energy level. We generally arrange orbitals by their energy level. To be able to solve the Schroedinger equation, you would need to study partial differential equations. Fortunately for us, lots of people have already solved this equation so we can just look up their solutions. The figure below shows the first 16 solutions to the Schrodinger equation for the hydrogen atom.

The red sphere, called an s orbial, is the orbital corresponding to the lowest energy level and this is where the electron in hydrogen starts. If the electron in the red sphere gains energy, it will move up to one of the higher energy levels (the yellow p, blue d, or green f orbitals).

That's really complicated, and the answer probably isn't what your book says.

The "normal" way of drawing this is to say that the atoms are in orbit around the nuclei the same way the Moon orbits the Earth and the Earth orbits the Sun, but this is wrong (see below). Rather, electrons form clouds of probability around the nuclei such that at some particular point in space, the electron is both partially there and partially not there, and if you add up all of the "partially there's" around the nucleus, they would sum to the total number of electrons around the atom. This is because of quantum mechanics:

"a particle as small as an electron does not have a single location in space where it is located, and cannot be forced into or even defined in a single, coherent manner"

If this seems difficult for you to believe, then now you understand why quantum mechanics is so hard for many people (Albert Einstein included) to swallow.

The reason electrons cannot simply orbit the nuclei is because electrons emit light as they are accelerated, and anything orbiting anything is constantly being accelerated (the mis-named "centrifugal" force). Emitting this light causes the electrons to lose energy, and spiral inward into whatever they are orbiting. For electrons, they would have to be orbiting so close that they would emit all their energy and crash into the nucleus in a tiny fraction of a second. Obviously, because atoms exist and are fairly stable in the universe, this cannot happen, so some other, more exotic, way to describe the way that electrons interact with atomic nuclei needed to be found.

Atoms are composed of three sub-atomic particles -- protons and neutrons at the center, the nucleus, and electrons outside of the nucleus. The protons have a positive charge the electrons have a negative charge and the neutrons are neutral. The electrons are attracted to the nucleus by the electrostatic force of attraction to the protons.

Surprisingly enough, although the minimum energy classically would have the electrons all located at the nucleus (point of strongest attraction) they aren't all there. The laws of quantum mechanics force the electrons to occupy particular positions around the nucleus, called orbitals, that depend on their total kinetic and potential energy. It is not at all obvious that this should happen; the explanation was determined by Werner Heisenberg and Wolfgang Pauli in the late 1920s.

There are some nice pictures and animations of orbitals, at: