In terms of engineering and material science are many many types of hardness, strength, and toughness.
Basic Idea:
Strength is the amount of _force_ required to break a sample.
Toughness is the amount of _energy_ required to break a sample.
Hardness is the size of an indentation when a pointy tip of specific shape and hardness is pressed against the material at a specific force.
Since force is not the same as energy, it follows that strength is not the same as toughness.
Types of strength:
Tensile strength: How much force it takes to break a sample by pulling. Units are typically force per area.
Compressive strength: How much force it takes to break a sample by pushing. Units are typically force per area.
Sheer Strength: How much force it takes to break a sample by a sideways (sheering) force. Units are typically force per area.
Yield Strength: How much force a sample can take before it deforms inelastically.
There are other types of strength as well.
Types of Toughness:
Impact Toughness: Typically measured by a Charpy Impact test: A pendulum (of standard shape, weight length) is swung into a sample (often a bar with a notch), and the energy adsorbed by breaking the sample is measured. Typical units are energy per area.
Tensile Toughness: A sample is slowly stretched until it breaks. The energy used to do this is an integral; namely the area under the stress vs. strain curve. Stress = force, strain = displacement. Energy = Work = force*dispacement = Stress*Strain = an integral. Units are in energy (Joules, foot-pounds, etc.)
Fracture Toughness: Energy required to grow a crack in a material. Units are energy per area.
There are other types of Toughness as well.
Hardness:
Hardness is more or less defined procedurally. Almost all hardness tests (Vicker's, Rockwell, Brinell, etc.) are a test where a pointy diamond is pressed into a sample at a fixed force and/or fixed time. The diamond must have a specific shape (sphere, rounded cone, pyramid, etc.). Sometimes a tungsten carbide indenter is used instead of diamond, but I think diamond is the most common. There are many many types of hardness tests, and they all use different parameters. In the end they have some measure of "size" of the indentation. For Rockwell C Hardness, the "size" is the depth of the indentation. So, informally speaking, the units for Rockwell C Hardness is length: Each point of HRC represents 2 microns of penetration. For Vicker's Hardness, the "size" is the area of the indentation. In Vicker's hardness, the units are force/area (similar to pressure, although the exact meaning is not quite the same).
If you want to know more about any specifics, you can find most of these on wikipedia.
Plus, I really like the the following references:
(1) Best popular science introduction to material science, in my opinion:
_Why Things Break_ by Mark E. Eberhart (2004)
http://www.amazon.com/gp/product/14...&pf_rd_t=101&pf_rd_p=470938631&pf_rd_i=507846
(2) Another good popular science introduction to material science. A little old fashioned, but very good, and more detailed than (1).
_The New Science of Strong Materials_ by J. E. Gordon (2006)
http://www.amazon.com/Science-Materials-through-Princeton-Library/dp/0691125481/ref=pd_bxgy_b_text_c
(3) I think this is a nice tutorial web-page on several times of hardness types (Rockwell, Brinell, Vickers, etc.):
http://www.gordonengland.co.uk/hardness/
Hope this helps!
By the way when people say,"strength is proportional to hardness" they mean Vicker's Hardness (and similar types), but not Rockwell Hardness. The Rockwell Hardness test is actually rather strange: An _infinitely_ hard material will have a finite Rockwell C Hardness of 100 HRC. This is clear if you go through the description of Rockwell Hardness from (3).
If you read (1) or (2), you'll have some idea as to why hardness and toughness are usually traded off. This trade-off is not always there, but is fairly common.
