Some stars are not steady. A Cepheid swells and shrinks on a fixed cycle of days to weeks, and its brightness rises and falls with it. The rise is fast and the fall is slow, and the pattern repeats so exactly that the period can be timed to a fraction of a day.
Henrietta Leavitt measured over a thousand of them in the Small Magellanic Cloud. Everything in that cloud is at essentially the same distance from us, so differences in apparent brightness there are differences in real brightness, with the distance divided out.
What she found was a clean relation: the longer the period, the brighter the star. A Cepheid pulsing every 30 days is far brighter than one pulsing every 3.
That turns a Cepheid into a measuring rod. The period is read off a clock and owes nothing to distance. The period gives the absolute magnitude, the observed brightness gives the apparent magnitude, and the gap between them is the distance modulus. Cepheids are bright enough to be picked out in other galaxies, so this reaches far past parallax.
Parallax works out to about 1000 parsecs. Beyond that, fitting a star to the main sequence carries the measurement to around 10^{5} parsecs, which covers our own galaxy and its neighbours.
Cepheids take over from about 10^{4} parsecs and reach some tens of millions. Past that they are too faint to pick out, and the last rung is the type Ia supernova.
A type Ia is a white dwarf that has taken on mass from a companion until it crossed the Chandrasekhar limit. Since every one of them detonates at the same mass, every one reaches close to the same luminosity. They are bright enough to be seen billions of parsecs away, and knowing their luminosity is what makes them useful.
Every rung is calibrated against the one below it, in the region where the two overlap. That is the strength of the arrangement and also its weakness: an error low down propagates all the way to the top. Much of the long argument over the value of the Hubble constant has been an argument about the lower rungs.
Take the spectrum of a distant galaxy and the absorption lines are all there, in the same pattern and the same relative spacing, but the whole pattern sits at longer wavelengths than it should.
The size of the shift is written z = \dfrac{\lambda - \lambda_0}{\lambda_0}, where \lambda_0 is the wavelength the line has at rest. While z is small it is close to v / c, the speed as a fraction of the speed of light.
Vesto Slipher measured these shifts from 1912. Nearly every galaxy he looked at was shifted to the red, and the further away it was, the larger the shift.
It is worth being careful about what the shift is. For nearby objects it really is the Doppler effect. For distant galaxies it is not: the light was stretched while it was in flight, because the space it was crossing grew while it crossed.
Edwin Hubble put the two measurements side by side in 1929: distances from Cepheids, speeds from redshifts. The points fell on a straight line through the origin. Georges Lemaître had derived the same relation from general relativity two years earlier.
The law is v = H_0 d, and H_0 is the Hubble constant. The current value is about 70 kilometres per second for every megaparsec of distance.
The constant is not dimensionless. Divide one by it and the units come out as a time, and \dfrac{1}{H_0} works out at around 14 billion years. That is a rough measure of how long the expansion has been going on, and it lands remarkably close to the age worked out by other means.
The value is still not settled. Measurements built on the distance ladder and measurements built on the early universe give answers that differ by more than either claims as its error. The disagreement is real and unresolved.
If everything is moving away from us, it is tempting to conclude that we are at the center. But the same law seen from any other galaxy gives the same picture: everything receding, speed proportional to distance. There is no point that the expansion is happening away from.
That falls out of the law itself. If every separation grows by the same fraction in the same time, then a galaxy twice as far recedes twice as fast, whichever galaxy you measure from. Uniform stretching looks the same from every point on it.
So the galaxies are not travelling through space. The space between them is getting longer, and they are carried apart by it. Nothing has to move for the distance to grow.
One consequence is that a galaxy far enough away recedes faster than light, which is allowed: the limit applies to motion through space, and this is not that. The other consequence runs backwards. If everything has been separating, everything was once together, and that is where the story of the first three minutes came from.