Why are wavelet transforms (Multi Resolution Analysis) used more in practice for compression rather than Fourier series?
In short: wavelets and relatives are pretty good at compacting a sufficiently large class of regular-enough and useful signals and images. What follows are signal properties (and corresponding wavelet features) that make wavelets good (not best) candidates for lossy compression:


piecewise-smooth (vanishing moments, or cancellation of low-order polynomials)
edges or jumps (gradient-like or Laplacian behavior of wavelets)
localized oscillations (zero-average and wiggling wavelet shape)
noise or spurious events (orthogonality and sparsity enhancement)


As said by @hops, the efficiency of wavelets for compression depends of the good matching of a signal class and the chosen wavelets.

Let us  restrict here to non-redundant discrete transformations: discrete Fourier and discrete wavelets. Both are orthogonal, or close enough (bio-orthogonal wavelets) to skip the distinction. 

So both, when transform coefficients are discarded, are least-squares approximations. But least-squares are not the best metric for compression: if you double the amplitude of a sample, the energy is multiplied by $4$, but it only adds one bit to the stored data, in a $\log_2$ reasoning.

In a way, a transform will help compression if it reduces the logarithmic cost or bit-budget of a discrete signal; hence, if the coefficients have a power law $1/c^{\beta}$ that decreases fast enough (the highest the $\beta$, the better).

This feature is often called the compressibility of signals. It can be assessed empirically, but also theoretically based on complicated functional analysis (Besov, Sobolev spaces). 

However, consider the useful class of $C^\alpha$ piecewise regular signals, $\alpha \ge 1$. They are locally smooth, with jumps (edges). Taking the largest $M$ coefficients for compression (or nonlinear approximation), the mean squared error for the Fourier basis will asymptotically decrease in $1/M$ for Fourier (because of Gibbs ringing). While wavelet approximations can decrease as $1/M^{-2\alpha}$. The smoother the signal, the better the wavelet compressibility. In  other words, Fourier cannot use the regularity of the signal.

In practice, this is not so simple. The complex aspect of Fourier is somewhat difficult to handle.  Quantization should be taken into account. The storage of the highest coefficients location costs too. Perceptual distortion should play a role, at least for lossy compression. 

So at low compression rates, local Fourier decompositions like the JPEG DCT can perform as well as wavelets, since asymptotic proofs do not apply. Indeed, local cosine bases, lapped orthogonal transforms that bridge the gap between Fourier and standard wavelets perform well too, and they are used in the MP3 standard (MDCT). For images, the 2D aspect renders separable wavelets less efficient than in 1D.

For strict lossless compression through, (wavelet) transforms are not the state-of-the-art yet.