What would occur if an area of land were to receive significantly less rainfall than usual for an extended time?

It really depends on how much less rainfall it it receiving and how long 'extended' refers to.

If we're taking those terms to their logical extremes, we can infer that the plant life in the area which isn't adjusted to a lack of rainfall will start to wither and die because they no longer have the necessary amount of water to live, and lack the evolutionary traits which would allow them to thrive in these conditions.

This would cause a domino affect, first driving away/killing off the primary consumers (herbivores) and slowly doing the same to all the other animals further up the food chain.

But again, this is taking the terms 'significant' and 'extended' to a logical extreme. In most cases, a process like this would take a large amount of time, and usually wouldn't result in a complete ecosystem collapse unless the climate was to undergo a complete shift.