Malleability: ability to be deformed (without fracture).
Ductility: ability to be drawn through a die (as in a wire drawing die)
Elongation: ability to be stretched under a tensile load
Tensile strength: ability to sustain a tensile load (pulling force)
Elasticity: ability to return to the original length after a force has been removed (could be tensile or compressive force).
Toughness: ability to absorb energy. There is dynamic toughness which is measured by the Charpy and Izod tests, and there is static toughness which is the area under the stress strain curve as determined by a reduced section tensile test.
Hardness: the resistance to indentation.
Hard materials are strong materials. The relationship between hardness and tensile strength is a direct relationship, i.e., as the hardness goes up, tensile strength goes up, but the relationship between hardness and ductility is an inverse relationship, i.e., as hardness goes up, ductility goes down. So, As hardness and strength goes up, ductility goes down.
A ductile material typically has good toughness, that is, a ductile material will absorb more energy and fail in a ductile manner, i.e., it displays visible distress before failure. Whereas a hard material that is strong may fail in a brittle manner in that it will not absorb much energy before failing without visible signs of distress. While a malleable material such as lead will not absorb much energy because it isn't ductile as copper or iron in the metallurgical sense.
These are paraphrased definitions as I remember them. I'm on the road and I can't look them up, but you should be able to google the terms as easily as I can. With my memory, I would advise you to google them. ;)
Best regards -Al