I can find sources that tell me explicitly what an NDR-pair is in terms of a map and a homotopy, but is there a good intuitive idea corresponding to, or some canonical examples that I might be able to investigate?
I found this article to be helpful: http://amathew.wordpress.com/2010/10/08/examples-of-cofibrations/
Basically, $(X, A)$ is an NDR-pair when there is an open $U \supset A$ (a neighborhood of $A$) such that $A$ is a deformation-retract of $U$. If $A$ is closed in $X$, then this is equivalent to the inclusion $A \hookrightarrow X$ being a cofibration.
I believe the most useful and ubiquitous examples (to an algebraic topologist) are CW-pairs $(X, A)$.