What is the sparse Fourier transform?
The idea of the algorithm is this: assume you have a length $N$ signal that is sparse in the frequency domain. This means that if you were to calculate its discrete Fourier transform, there would be a small number of outputs $k \ll N$ that are nonzero; the other $N-k$ are negligible. One way of getting at the $k$ outputs that you want is to use the FFT on the entire sequence, then select the $k$ nonzero values.

The sparse Fourier transform algorithm presented here is a technique for calculating those $k$ outputs with lower complexity than the FFT-based method. Essentially, because $N-k$ outputs are zero, you can save some effort by taking shortcuts inside the algorithm to not even generate those result values. While the FFT has a complexity of $O(n \log n)$, the sparse algorithm has a potentially-lower complexity of $O(k \log n)$ for the sparse-spectrum case. 

For the more general case, where the spectrum is "kind of sparse" but there are more than $k$ nonzero values (e.g. for a number of tones embedded in noise), they present a variation of the algorithm that estimates the $k$ largest outputs, with a time complexity of $O(k \log n \log \frac{n}{k})$, which could also be less complex than the FFT.

According to one graph of their results (reproduced in the image below), the crossover point for improved performance with respect to FFTW (an optimized FFT library, made by some other guys at MIT) is around the point where only $\frac{1}{2^{11}}$-th to $\frac{1}{2^{10}}$-th of the output transform coefficients are nonzero. Also, in this presentation they indicate that the sparse algorithm provides better performance when $\frac{N}{k} \in [2000, 10^6]$. 



These conditions do limit the applicability of the algorithm to cases where you know there are likely to be few significantly-large peaks in a signal's spectrum. One example that they cite on their Web site is that on average, 8-by-8 blocks of pixels often used in image and video compression  are almost 90% sparse in the frequency domain and thus could benefit from an algorithm that exploited that property. That level of sparsity doesn't seem to square with the application space for this particular algorithm, so it may just be an illustrative example.

I need to read through the literature a bit more to get a better feel for how practical such a technique is for use on real-world problems, but for certain classes of applications, it could be a fit.