Anyhow, as an algebraic geometer, the place I learned about Uhlenbeck's work was in compactification of moduli of curves. Basically, say I want to look at all the polynomial curves that can live in a certain 'environment'. (One 'application' is string theory -- what particle interactions can occur given a certain set of energy constraints? The particle interactions are curves traced out over time by particles/Riemann surfaces traced out over time by the little loops that represent closed strings in string theory; the energy constraints constrain the shapes the particle interaction can take.) If you want to look at limits of families of these interactions, you're looking at some odd behavior. An example is the equation xy = t^2 -- this gives you nice hyperbolas for values of t not equal to zero, but as t -> 0, you get something singular, two crossed lines. Or x^2 - y^2 - z^2 = t^2: you have a nice smooth hyperboloid when t neq 0, and a double cone when t=0. However, these examples are just showing you the idea of how singularities appear in families of smooth polynomials. Uhlenbeck's older work really worked with the system of constraints that I mentioned -- figuring out what can appear under those constraints is a considerably more complicated problem!
This is a very imprecise discussion above and I'm totally blurring together Uhlenbeck's bubbling and compactness theorems from gauge theory -- but I never followed her work in a systematic way but instead was plunged into seeing its aftereffects from another related field that the work affected.
Last quote really resonates: “Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness, and self-absorption. One cannot ask for more.”