(Image from Wikipedia.)
First of all, most of us learn about relativity and think that it's only applicable at insanely high speeds or in bizarrely high gravitational fields (i.e. black holes, astrophysics and stuff like that). Not so! I'll list just a few examples here, but there are lots of cool places where relativity pops up in every day life.
Global Positioning System (GPS)
Unless you happen to have been living in a cave for the last decade, you've probably had the chance to play with a GPS receiver. The physics behind how the whole system works is actually really interesting. When you think about the fact that you can stand on the street corner with a device not much bigger than your cell phone (if it's not already in your cell phone) and tell your position on the globe to within a few feet, that's amazing!
Anyways, the way GPS works is really just a matter of clocks and light. We know the speed of light very precisely. If you take about 24 satellites, and load them each with a transmitter and some of the world's most precise time pieces, you really have just about all you need for GPS. From simple geometry, if you know the distance you are from four satellites, you know your position on the earth. From basic physics, if you receive a signal from a satellite and you know how long ago the signal was sent, then you know your distance from the satellite (d = c t). (It actually gets even easier than this. The problem is basically three equations with three unknowns for position. However, if we have a fourth satellite, we can make it four equations with four unknowns for position and time. So your GPS receiver doesn't even need a very good clock and it can give you the time to atomic clock precision!)
Now, as you can see, the whole system revolves around a very precise understanding of time. Also, since the speed of light is just about exactly 1 foot per nanosecond, you don't have to have a very big error in your time measurements to be way off in your position measurement. Now, if we are going to understand time, we have to have a very good theory about how time works, right? What's the modern theory about time -- Relativity!
It turns out that relativity isn't just nice to have to make GPS better. It's not even just to give us the extra precision to go from "plus or minus a few meters" down to "plus or minus a few centimeters." Without correcting for relativity, you would very quickly be off by hundreds of kilometers! (FYI, As I understand, in this case, the corrections due to General Relativity are actually larger than the corrections due to Special Relativity.)
During the summers, I help out with a teacher development program here at UIUC. One of the things we talk about (as an excuse to talk about a whole lot of cool physics) is GPS. A friend of mine, Kevin Zielnicki, wrote a cool program showing the effects that things like clock uncertainty and correcting for relativity would have on GPS accuracy. The program is available here. You can play with it and see how quickly your "apparent position" gets off from your "actual position" (set to be at Loomis Lab on UIUC campus :) ). Don't worry so much about the "spikes" in the uncertainty plot -- those are mainly due to when the visible satellites all end up co-planar and your equations are no longer independent. Anyway it's really interesting to see how quickly your position gets off if you don't correct for relativity. (Don't ask me how he wrote this program. All I know is he's a whole lot better at this sort of thing than I am, and he spent half a semester coding everything up. Also, if you go poking around there are a bunch of other cool lesson plans and stuff on that website, if you are into that sort of thing.)
The point is, without Relativity, the Global Positioning System would not even be possible, let alone be as insanely accurate as it is!
Believe it or not, every time you play with your favorite refrigerator magnet, you are demonstrating Special Relativity.
Special Relativity not only mixes space and time into spacetime, but it also mixes electric fields and magnetic fields into the Electromagnetic Tensor.
Most of the headaches in learning relativity come from wrapping your mind around the idea that space and time are not really the separate entities we usually think of them as, but rather they are really better viewed as a combined spacetime. In other words, measurements that one observer would conclude are due to a difference in location, or space, another observer would accurately conclude are due to a difference in time. This is what leads to problems with the ideas of simultaneity, absolute length, etc. Similarly, what one observer measures as a purely magnetic field, another observer would measure to be a purely electric field, or a mixture of the two. In one sense, magnetic fields are simply due to viewing electric fields from a moving frame.
Now you might think that you'd have to move pretty fast to view an electric field to be a magnetic field. However, this is not the case. For example, the electrons in a standard copper wire of cross-sectional area 7 mm2 (about 3 mm diameter) carrying 1 A of current are only moving about 0.01 mm/s. Yet this is fast enough to have relativistic effects!
Why Mercury is Liquid and Gold is Yellow
First of all, a disclaimer: I am not a chemist. I work for a chemist, and I took a bunch of chemistry classes a long time ago, but I am not a chemist, and in this area I'm really not the best person to ask. However, I do know enough to know that this stuff is really cool, and it's amazing how something like Relativity can come into answering fairly simple questions like "Why is Gold yellow?" or "Why is Mercury a liquid?" (Just like Rayleigh Scattering explaining "Why is the sky blue?") Just about all of my information on this comes from sources here and here.
Now, it turns out that Relativity comes in to affecting the chemistry of a lot of the heavier elements. The big reason behind this is that as objects move faster, their masses increase. (Note: You may argue about whether it's better to think about the mass increasing or the energy / momentum formula changing (or later if it's ok to think about electrons "moving" inside an atom), but for the purpose of this explanation, we'll take the view of a relativistically moving electron having an increasing effective mass.) Now, how fast are the electrons effectively moving in an atom? For the lighter elements, not very fast. For example, for the 1s electron in Hydrogen, γ is only about 1.00003. (Note that even so, Sommerfeld took these relativistic corrections into account in his 1916 refinement of Bohr's model.) Now, for larger atoms, like Gold (Z = 79) and Mercury (Z = 80), the electrons can be going really fast (for the 1s electrons, γ would be about 1.22 and 1.23, respectively). In general, the relativistic corrections mean that the effective orbital radius for some orbitals contracts a lot, but for others, it only changes a little. This results in a splitting of energy levels.
Now, we can talk about some of the physical and chemical properties we alluded to earlier. The graph on the right (from here) shows the relevant energy levels for Silver and for Gold (actually diatomic molecules AgH and AuH, but it works the same way). These were calculated both accounting for relativity (rel.) and not accounting for relativity (n.r.). The reason Gold is yellow is because the 5d electrons can absorb blue light and transition to the 6s level. The corresponding transition in Silver, on the other hand, is in the UV, so Silver is colorless. Note that this only works using the relativistic corrections to the energy levels. To understand why Gold is yellow, we need Relativity! That's cool!
Now, these effects lead Gold (Z = 79) to have a very high electron affinity -- so much so that it has been called a "pseudo-halogen." In fact, in certain compounds it acts a lot like a halogen like Iodine. For the same reason, its neighbor Mercury (Z = 80) acts a lot like a noble gas. The filled 6s2 orbital is so contracted (due to Relativity), that it doesn't really contribute to the Hg-Hg bonds. Just like in Helium, the filled 1s2 don't contribute to bonding, so there is no He2. Such a bond would have to fill the excited anti-bonding levels. (However, Gold acts much more like Hydrogen because the unfilled 6s1 electron can form good Au-Au bonds -- just like in H2 molecules!) So, Mercury has a low melting point for the same reason that Helium or any of the noble gasses do -- it has really weak bonds with itself -- all due to Relativity!
Well, that's just a taste of where we see Relativity beyond black holes and astrophysics. It's really amazing to see how much influence it has.