Elasticity is how much you can deform (stretch) an object, before the deformation is no longer reversible. Think of a rubber band, you can stretch it a little, and it returns to its original shape. Stretch it a lot, and it breaks down and does not return to its original shape.
Plasticity is how much you can deform an object before it breaks. This is the snapping point in the rubber band, where it completely breaks.
These two properties are different, and have to do with how the material is deforming on an atomic level. During elastic deformation, the individual bonds stretch. During plastic deformation, regions, layers, or fibers of the material slip. The relationship is best described by the stress-strain curve [1]. This plots deformation versus the pressure required.
Hardness and Toughness are the compression version of Elasticity and Plasticity.
These properties are often found experimentally, because they are a summation of many different atomic level interactions. Infact most materials are anisotropic, meaning if I measured the toughness of a material along two axises, I would get two different values.
[1] https://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve