When learning about atoms, students soon learn about isotopes atoms of the same element that vary only in how many neutrons they have.
When learning about elements, the concept of allotropes is similar. Allotropes are different forms of an element that vary only in how the atoms are connected.
Carbon is a familiar element, with allotropes you are well aware of, though you may not have paused to think about it. Graphite (used in pencil lead and some lubricants) is a structure in which the carbon atoms are arranged in hexagons, forming single layer sheets. (These sheets slide easily over one another, which is why graphite can make a good lubricant.) Diamond is also carbon, but now the structure is a three-dimensional one, forming a strong network of carbon atoms in a tetrahedral arrangement. There is also the synthetic buckminsterfullerene, another allotrope of carbon where sixty carbon atoms are arranged to form a soccer-ball like frame, leading to the nickname "buckyball". Other synthetic forms of carbon are now made, but one is enough for this illustration.
Oxygen also comes in two allotropic flavors oxygen and ozone. Plain diatomic oxygen is two oxygen atoms bound together (by a double bond). Ozone is three oxygen atoms in a bent arrangement. It has a single and a double bond that swap back and forth through resonance, or, if you'd prefer things a little more advanced, it has a delocalized electron pair. For our purposes, the key here is that oxygen and ozone are structurally distinct.
By now, you get the idea, but I have one more example for you. The previous two are the most familiar, but here is one you can actually have some fun with.
Sulfur has several allotropes. The common form of sulfur is "crown sulfur", so called because eight sulfur atoms are arranged in a zigzagging loop that looks a little like a crown, having four points up and four points down. (You might recognize that jagged circles are not well-suited to making larger structures, which is why sulfur crumbles so easily.) The bonds that hold the sulfur rings together are not very strong, and can be broken by heating gently.
[If you are trying this, make sure you wear goggles and work in a ventilated area. Also be careful not to ignite the sulfur, as it produces toxic sulfur dioxide, which will burn eyes, nose, throat, and lungs.]
The sulfur melts, and then turns a red color as the crowns break open, forming eight atom chains of sulfur. Those chains are then free to join, end to end to make longer chains. Once the sulfur has changed color, you can allow it to cool either on its own, or cool it quickly by pouring it into water. The result is a different allotrope of sulfur called "plastic sulfur", for obvious reasons. The longer chains of sulfur are tangled with one another, and the product looks very much like a lump of yellow plastic.