Updated 2017 by Michael Weiss (current Hubble constant).
Original by Michael Weiss.
Mrs Felix: Why don't you do your homework?
Allen Felix: The Universe is expanding. Everything will fall apart, and we'll all die. What's
the point?
Mrs Felix: We live in Brooklyn. Brooklyn is not expanding! Go do your homework.
(from Annie Hall by Woody Allen)
Mrs Felix is right. Neither Brooklyn, nor its atoms, nor the solar system, nor even the galaxy, is expanding. The Universe expands (according to standard cosmological models) only when averaged over a very large scale.
The phrase "expansion of the Universe" refers both to experimental observation and to theoretical cosmological models. Let's look at them one at a time, starting with the observations.
The observation is Hubble's redshift law.
In 1929, Hubble reported that the light from distant galaxies is redshifted. If you interpret this redshift as a Doppler shift, then the galaxies are receding according to the law:
speed of recession = H × distance from Earth.
H is called Hubble's constant; Hubble's original value for H was 550 km/s/Mpc (which is about 169 km/s/Mly, where "Mpc" means mega parsec, and Mly means mega light-year: see next paragraph). The latest estimates of H put it at 69 km/s/Mpc (21 km/s/Mly; see this NASA web page, or this page from the Harvard-Smithsonian Center for Astrophysics for more details.)
A word on units. Astronomers almost always report H in kilometres per second per mega parsec (km/s/Mpc). The parsec is a traditional unit of distance, equal to about 3.26 light-years. Neither the parsec nor the light-year is an official SI unit, although both are sanctioned by the IAU. But there is (for my money) a more illuminating way to express the constant. Note that km/Mpc is dimensionless—its dimensions are length/length! So the SI unit for H is really the inverse second, s−1. This means that 1/H is a time. You might guess it's the age of the universe. It is a rough estimate for that age, but because H changes with time (see below), the actual age is different. The present-day value of H is written H0. So, using H0 = 21 km/s/Mly, we estimate 1/H0 = 1/21 Mly/km s = 14.2 thousand million years. This is a bit larger the current best estimate, 13.9 thousand million years. Hubble's original value, 169 km/s/Mly, translates to about 1.8 thousand million years.
Hubble's redshift formula does not imply that Earth is in particularly bad odor in the universe. The familiar model of the universe as an expanding balloon speckled with galaxies shows that Hubble's alter ego on any other galaxy would make the same observation.
But astronomical objects in our neck of the woods—our solar system, our galaxy, nearby galaxies—show no such Hubble redshifts. Nearby stars and galaxies do show motion with respect to Earth (known as "peculiar velocities"), but this does not look like the "Hubble flow" that is seen for distant galaxies. For example, the Andromeda galaxy shows blueshift instead of redshift. So the verdict of observation is: our galaxy is not expanding.
The theoretical models are, typically, Friedmann–Robertson–Walker (FRW) spacetimes.
Cosmologists model the universe using "spacetimes", that is to say, solutions to the field equations of Einstein's theory of general relativity. The Russian mathematician Alexander Friedmann discovered an important class of global solutions in 1923. The familiar image of the universe as an expanding balloon speckled with galaxies is a "movie version" of one of Friedmann's solutions. Robertson and Walker later extended Friedmann's work, and so you'll find references to "FRW spacetimes" in the literature.
FRW spacetimes come in a great variety of styles—expanding, contracting, flat, curved, open, closed, ... The "expanding balloon" picture corresponds to just a few of these.
A concept called the metric plays a starring role in general relativity. The metric encodes a lot of information; the part we care about (for this FAQ entry) is distances between objects. In an FRW expanding universe, the distance between any two "points on the balloon" does increase over time. But the FRW model is not meant to describe our spacetime accurately on a small scale—where "small" is interpreted pretty liberally!
You can picture this in a couple of ways. You may want to think of the "continuum approximation" in fluid dynamics—by averaging the motion of individual molecules over a large enough scale, you obtain a continuous flow. (Droplets can condense even as a gas expands.) Similarly, it is generally believed that if we average the actual metric of the universe over a large enough scale, we'll get an FRW spacetime.
Or you may want to alter your picture of the "expanding balloon". The galaxies are not just painted on, but form part of the substance of the balloon (poetically speaking), and locally affect its "elasticity".
The FRW spacetimes ignore these small-scale variations. Think of a uniformly elastic balloon, with the galaxies modelled as mere points. "Points on the balloon" correspond to a mathematical concept known as a comoving geodesic. Any two comoving geodesics drift apart over time, in an expanding FRW spacetime.
At the scale of the Solar System, we get a pretty good approximation to the spacetime metric by using another solution to Einstein's equations, known as the Schwarzschild metric. Using evocative but dubious terminology, we can say this models the gravitational field of the Sun. (Dubious because what does "gravitational field" mean in GR, if it's not just a synonym for "metric"?) The geodesics in the Schwarzschild metric do not display the "drifting apart" behaviour typical of the FRW comoving geodesics—or in more familiar terms, Earth is not drifting away from the Sun.
By the way, Hubble's constant, is not, in spite of its name, constant in time. It varies in a non-uniform manner. We might well have two competing effects. On the one hand, the overall mass of the universe tends to slow the expansion over time. On the other hand, any mysterious dark energy might accelerate it. Actually, the term "dark energy" is just a placeholder for a theory to come: it's a way of talking about the fact that the expansion is currently thought to be accelerating (a deduction of some sort from observations). We don't understand why the expansion might be accelerating. Although not constant in time, H is (in the FRW models, with certain caveats) constant over space at any given time.
The "true metric" of the universe is, of course, fantastically complicated; you can't expect idealized simple solutions (like the FRW and Schwarzschild metrics) to capture all the complexity.
In newtonian terms, we say that the Solar System is "gravitationally bound" (ditto our galaxy, and even the "Local Group" of galaxies). So the Solar System is not expanding. The case for Brooklyn is even clearer: it is bound by atomic forces, and its atoms do not typically follow geodesics. So Brooklyn is not expanding. Now go do your homework.
(My thanks to Jarle Brinchmann, who helped with this list.)
Misner, Thorne, and Wheeler, Gravitation, chapters 27 and 29. Page 719 discusses this very question; Box 29.4 outlines the "cosmic distance ladder" and the difficulty of measuring cosmic distances; Box 29.5 presents Hubble's work. MTW refer to Noerdlinger and Petrosian, Ap. J., 168 1–9 (1971), for an exact mathematical treatment of gravitationally bound systems in an expanding universe.
M.V.Berry, Principles of Cosmology and Gravitation. Chapter 2 discusses the cosmic distance ladder; chapters 6 and 7 explain FRW spacetimes.
Steven Weinberg, The First Three Minutes, chapter 2. A non-technical treatment.
Hubble's original paper: A Relation Between Distance And Radial Velocity Among Extra-Galactic Nebulae, Proc. Natl. Acad. Sci. 15, No. 3, 168–173, March 1929.
Sidney van den Bergh, The cosmic distance scale, Astronomy & Astrophysics Review 1989 (1) 111–139.
M. Rowan-Robinson, The Cosmological Distance Ladder, Freeman.
Two books may be consulted for what is known (or believed) about the large-scale structure of the universe:
P.J.E. Peebles, An Introduction to Physical Cosmology.
T. Padmanabhan, Structure Formation in the Universe.