3d printers are machines which convert digital models transferred from a computer into solid objects which can be view and handled in the physical world.
They work by first slicing the digital model into 1000's of pixel thin cross sections. The 3d printer then "prints" a birds-eye-view copy of the very bottom cross-section of the model. Unlike your inkjet which uses ink a to re-create the cross section's shape onto a sheet of paper, the 3d printer "prints" a super-thin layer of solid material such as plastic or metal onto a print-bed. For the next cross-section up, the inkjet printer would feed the previous sheet of paper to the out-tray and then print the shape of the next cross-section onto a new sheet of paper. The 3d printer however would "print" the next cross section's shape right on top of the previous one. The 3d printer would then continue to print layer upon layer of each successive cross-section image until eventually, a completed, solid version of digital model is formed on the print bed.
3d printing is a bit like decorating a cake. When you decorate a cake you squeeze the soft icing / frosting out through a nozzle in the end of a bag to create three dimensional text or shapes. By moving your hand around while squeezing the bag, the icing will be laid on to the cake in whatever shape you want. To increase the height of the shape you would squeeze layer upon layer of icing onto the cake until the desired design is formed. Once the frosting is applied to the cake, it will begin to set until it is hard.
To create a 3d model, many 3d printers follow a similar process to cake decoration in that a mechanical nozzle squeezes a soft material onto a base plate to create objects. The nozzle or base plate moves around so that a layer of material can be laid onto the base plate to form the desired shape. Once a layer is applied, either the base plate moves down slightly or the nozzle moves up slightly so that a new layer of material can be applied on top of the previous one.
It may help your understanding of this process if you imagine a 3d model to be a loaf of sliced bread. If you were to take each slice of bread out of the bag and stack them on top of one another in the order you took them out of the bag, you would end up with a complete (precariously balanced, vertically stacked) loaf of bread. Each slice of bread is like a layer of material laid by a 3d printer. The slices of each layer of a 3d model however are very thin compared to a slice of bread. In fact to reach the thickness of a slice of bread, the 3d printer would need print 10,000 layers of material!
The process of adding successive layers of material on top of one another to form objects is known as Fused Deposition Modelling (FDM) and is just one of the technologies used on 3d printing. Other technologies include firing laser beams at powders or resins to produce solid layers and gluing layers of paper on top of one another. What all 3d printers have in common, is that they progressively add layers of material until the final object is formed. The proper name therefore which is given to this process is additive manufacturing. Since this is a bit of a mouthful, the term 3d printing is here to stay.