Graceful graphs arise in a number of contexts, but one well-known one is the Golomb ruler. The Golomb ruler can be described as a problem in radio astronomy: Using as few telescopes as possible (telescopes are expensive), construct a line of telescopes such that for every distance between 1 and N (for some N), there is a pair of telescopes that far apart from each other.
Each such arrangement corresponds to a Golomb ruler of length N. Such a ruler, when marked at only the telescope positions, could measure every distance from 1 to N. (Not every unit has to be marked, as on a traditional ruler.) When no distance is repeated, the arrangement is called a perfect Golomb ruler. Every perfect Golomb ruler of length N corresponds to a graceful labelling of the complete graph with N edges.
Graceful graphs can be thought of as natural generalizations of Solomon Golomb's famous "rulers," which I first read about years ago in a Scientific American article [Gar 1]. The basic premise is this: Suppose a radio astronomer needs to place satellite dishes along a straight line in such a way that every possible distance 1, 2, 3, ..., N is achieved between some pair of them. One way to accomplish this would be to put a satellite at position 0, then another at 1, then another at 2, and so on through N (thus using N+1 satellite dishes).
In general, of course, one wants to use as few satellite dishes as possible, and in the least amount of space. Obviously the above placing is not the best scheme possible. In fact, it shouldn't take you long to convince yourself that for N=6, the following works:
We could make this ruler into a graph by replacing each satellite dish with a vertex of the graph, and putting edges whenever the distance between two satellite dishes is used, as pictured above. Such a labelling is what Golomb called a graceful labelling of the graph, which is then said to be graceful. However, Golomb was not the first to study graceful labellings; Alexander Rosa studied them under the notion of alpha- and beta-valuations.