|The atom, with the central nucleus of protons and neutrons surrounded by an electron cloud.|
Protons and Neutrons comprise the center, or nucleus of atoms, while electrons buzz about in an "electron cloud" surrounding the nucleus. Now, this picture doesn't show it, but the electron cloud is very far away from the nucleus, and the electrons are so small and moving so fast we can't actually tell where they are at any given time.
To put in perspective how vast a distance there is between the center of an atom to the edge of its electron coud, if you put a pin in the very center of the 50 yard line on a football field, the electrons would orbit the field near the parking lot.
But why can't we know exactly where the electron is at any given point? Well actually, we can't truly know the location of any object in motion's location while also knowing its velocity. This is effectively described in Heisenberg's Uncertainty Principle.
For example, if we were to throw a baseball across a plate, we could use a radar gun to determine its speed, and take a picture at the exact same time we found the speed of the baseball. The picture would show the ball's location, but not its velocity, while the radar gun would show the ball's velocity but not it's location.
However, we can "see" the ball moving because the uncertainty of the position of the baseball moving across the plate is smaller than the radius of the baseball. This isn't true for electrons, however, which are so infintesimally small that the uncertainty of their position is greater than their radii, which means we cannot know their position at any given time, but we can estimate that they fall within a particular region of a cloud.
"Wait a second," you say, "I get why electrons orbit the positive nucleus (due to opposing electric charges encouraging attraction), but why is the nucleus sticking together?" This is due to the strong and weak nuclear forces, but also: the Neutron. The Neutron is neutrally charged, which means that it can be polarized by any charge, positive or negative. If a proton is brought close to it, they will attract because the neutron will polarize itself electrically (the positive charges in the neutron will move to the opposing side of the neutron, meaning the negative charges will be closest to the proton). The protons stick to the neutrons, which act like a glue to hold the nucleus together.
To demonstrate this principle, try this: Get two magnets. Like poles repel, opposite poles attract, just like electric charges in protons and electrons. So, if you put two North poles or two South poles together, they'll fly apart. However, put a piece of iron in between two like poles, and stick the magnets on different parts of the iron, You'll find you will now have one continuous piece of metal with the same poles facing inwards. The iron acts like a neutron, allowing itself to be temporarily polarized and stick the two like magnets together.