Let's say you have a normal (axial) elongation strain.  Look up your strain value on x axis of material's stress-strain curve to get corresponding y-axis value of stress-strain curve, which is the stress on material corresponding to that strain.  Conversely, if you have, say, a tensile stress value, look up your stress value on y axis of stress-strain curve and find corresponding x-axis value(s) of stress-strain curve, which is strain of the material corresponding to that applied stress.

Because strain (the x axis of stress-strain curve) is elongation (change in length) per unit length, and therefore a dimensionless percentage, it could also be referred to as strain "rate."

Notice, in paragraph 1, if the stress value you picked off of the stress-strain curve is closer to the origin than the so-called yield point, you are in the elastic range (if the material has one).  If you are further away from the origin than the yield point, you're in the plastic range.

If you are above the x axis on a stress-strain curve, the stress is tensile stress.  Below the x axis is compressive stress.  If a material has almost identical tensile and compressive stress-strain curves (above and below the x axis), they might only depict the tensile portion (above the x axis).

If the stress-strain curve has an initial linear (straight-line) portion, then stress can be computed according to Hooke's law, sigma = E*epsilon, where sigma = stress, E = normal modulus of elasticity (force per unit area), and epsilon = strain (dimensionless percentage).

The maximum tensile stress a material can support is called the ultimate tensile strength (UTS) and is the peak (highest y value) on the stress-strain curve.

If you have, say, a horizontal portion on a stress-strain curve (constant stress with a changing strain), it means the material is continuing to yield (elongate) even without adding more applied force.  In this situation (and let's assume the force is tension), if you do not remove the applied force, then the material will continue elongating either (a) until the strain reaches the point where the stress-strain curve begins to rise again, if the curve ever rises again, or (b) until the material reaches the rupture point (where the material snaps), if the curve never rises again.

For more information on stress-strain curves, elastic range, plastic range, strains, etc., perhaps try
http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture24/461_lecture24.html.
Good luck in your studies.

Correction.  "Strain rate" is strain per unit time.  This refers to the rate of stress application (e.g., quickly or slowly).

Stress-strain behavior is strongly strain-rate dependent.  Materials often respond very differently depending on the rate of strain.  In general, an increase in strain rate usually has the same effect as a decrease in temperature (i.e., an increase in modulus of elasticity but a decrease in strain to rupture).  For example, silly putty and the earth's mantle act as elastic solids when the strain is applied rapidly, but deform like a fluid when the strain is applied slowly.  High strain rates enter into most fracture, impact, erosion, and shock loading situations.

Conversely, however, certain materials have an ability to undergo extraordinary tensile deformation without fracturing under certain high strain rates, a property called superplasticity.  Materials that have demonstrated superplasticity include aluminum, titanium, nickel, steels, and even some ceramics.

To answer your question, to my knowledge there is no clear-cut relationship between ultimate strength and changing strain rate, since different materials respond very differently depending on the strain rate.