Why are rare earth metals important for electronics?
Although tantalum is not one of the rare earths -- it is one of the "transition metals", like gold -- tantalum's scarcity (1 or 2 ppm of the earth's crust) and primary use in electronics (tantalum capacitors) fits in with the scope of this question.  

Recent legislation in the US (July, 2010) requires companies to disclose if they are using products with tantalum obtained from the Democratic Republic of the Congo (DRC).  As a result, prices have risen sharply as other producers come slowly back on-line.  One mine in Australia represents 1/3 of the potential global world production.

  

(Note: vertical scale of the graph doesn't start at zero, it gives a slightly distorted look. Full size graph is here)

Because electrolytic tantalum capacitors can be much smaller than aluminum electrolytic capacitors of the same capacity, and have higher voltage ratings, they are used in almost all cell phones, and in other portable electronic equipment.

I designed in a couple of 1000 µF "tants" into a product a couple of years ago, and recently the contract manufacturer contacted us saying that the lead time on the parts had stretched out to 16 weeks and asked if I could find a substitute.  As a result of this exercise, in my latest design I went back to surface-mount aluminum electrolytic capacitors, even though there was a significant space penalty.