Good, Motivating Examples for Algorithmic Complexity
I attack this problem with three "visits" over the course of two years with my students


At our first brush with Complexity, I don't spend a ton of time motivating the study of it.  I introduce the notations of $\Omega$, $\Theta$, and $O$ (as well as briefly touching upon $\omega$, $\theta$, and $o$).  I show the difference between linear and binary search, and then explore four sorting algorithms: bubble, selection, insertion, and bogosort.  (This is fairly early on in the curriculum, and we haven't dealt with recursion yet, so I skip out on quicksort and mergesort.)  The last one makes them laugh, and makes the material at least have a point: some algorithms are just worse than others.
Later in the year, we look at quicksort and mergesort.  At this point, again, we use this only as a chance to review the concepts and the notation.
Near the beginning of the second year, I talk about why algorithmic study matters.  Computers get insanely faster over time, right?  (See Richard's answer for some eyebrow-raising numbers).  Since our computers are getting faster, it only stands to reason that things that are slow to compute today will also just get faster.

"Au contraire!" I cry, "as our computers get faster, our algorithms are getting slower, you see!"  What do I mean by this outrageous claim?  As our computers get faster, our hard drives also get bigger.  By necessity, we perform calculations with larger and larger data sets.    

So, let's use high-end rendering as an example.  In 1980, Apple came out with the Apple III computer, which shipped with a whopping 5 megabytes of diskspace.  It also used the 6502B processor.  It could perform about 0.43 million instructions per second (MIPS).  Today, hard drives come in at about 6 gigabytes, and 305 billion instructions per second.  My numbers don't have to be exact to make this work, though.

If we processed a complex graphic that took about .5% of our hard drive using a $O(n^3)$ algorithm in 1980, that would be about 2,500 bytes of data, and it would take us roughly 9.25 years to complete.  That's awful!!  But if we did that today, (roughly 30 megabytes and 305 billion computations per second), it would take us over 28 thousand years to complete.  We're trending the wrong way!  The problem is that, even at a relatively cheap function like $O(n^3)$, the growth due to data creep is just far faster than Moore's Law.  Faster computers can't deal with correspondingly bigger problems if the $O$ sucks.
Finally, a few weeks later, I talk in grand, futuristic terms about the Big Picture.  This is more to excite their curiosity than anything else. The three drivers of possibility with computers are, I argue, smaller transistors, better battery life, and better algorithms.  If you take a $O(n^3)$ algorithm and find a way to accomplish a similar task in $n^2$ time, that's a foundational enabler.  You don't become individually famous, but you change the world.  The Google Maps team didn't solve the Traveling Salesman problem, they developed algorithms that could get likely best answers without the guarantee.  The bottom line is that developing algorithms with better complexity literally enables us to do new things on practical timescales.  To paraphrase Heroes, change the algorithm, change the world.