The rate constant relates the reaction rate to the concentrations of the reaction components. 
Significance of the rate constant
The rate constant, k, gives a direct measure of the relative reaction rate.
A very small value for the rate constant equates to a very slow reaction in general. Equally, a large value for the rate constant means a large value for the rate and that the reaction is rapid.
Rate = k[A]^{x}[B]^{y} 
Example:

Clearly, the actual rate is dependent on the concentrations and qualified by the orders, but it is still possible to get an idea about the relative rate of a reaction just by looking at the rate constant.
Units of the rate constant
The units of the rate constant depend on the overall order of the reaction. As the rate equation can take different forms, i.e. the orders with respect to the individual reactants depend on the specific reaction under study, then the units for the rate constant change appropriately.
To find the units you must substitute the units of all of the other parts of the equation and then cancel down.
Units of rate = mol dm^{3} s^{1} Units of concentration = mol dm^{3} 
When the concentration is raised to a specific order then the same thing must be done to the units of concentration. i.e.:
[A]^{1} has units of (mol dm^{3})^{1} = mol dm^{3}
but [A]^{2} has units of (mol dm^{3})^{2} = mol^{2} dm^{6}
So, the rate constant k for a first order reaction rate = k [A]^{1} k = rate/[A]^{1} therefore units of k = mol dm^{3} s^{1}/mol dm^{3} = s^{1} 
For a second order reaction rate = k [A]^{2} k = rate/[A]^{2} therefore units of k = mol dm^{3} s^{1}/mol^{2} dm^{6} = dm^{3} mol^{1} s^{1} 
Note that the above refers to the overall order of the rate equation, not the specific individual orders. i.e the rate equation:
Rate = [A][B] 
is second order overall, therefore:
Rate constant (k) units = dm^{3} mol^{1} s^{1} 
Factors affecting the rate constant
The rate constant is just that, constant, unless the temperature conditions change. (Kinetics section 6.40)
However, catalysts also change the rate constant, but the reaction is now a different process with a different mechanism (Kinetics section 6.43).