Expanding Universe
The expanding universe is the observation that space itself is stretching, carrying galaxies apart, with more distant galaxies receding faster.
Definition
The universe is getting bigger: the space between galaxies is stretching, so distant galaxies move away from us, and the farther away a galaxy is, the faster it recedes. Edwin Hubble found in 1929 that a galaxy's speed away from us is proportional to its distance, a relation now called Hubble's law, and this expansion is the main evidence for the Big Bang. Formally, expansion is the growth of the scale factor $a(t)$ in the Friedmann-Lemaitre-Robertson-Walker metric, which multiplies all cosmological distances, with the present rate given by the Hubble constant $H_0 \approx 70\ \text{km/s/Mpc}$ and recession velocity $v = H_0 d$. Observations of Type Ia supernovae in 1998 showed the expansion is accelerating, an effect attributed to dark energy.
Example
Raisins in rising bread dough all move apart as the dough expands, with no raisin at the center; galaxies behave like the raisins and space like the dough. A galaxy $100$ million light-years away recedes at roughly $2{,}000\ \text{km/s}$, and one twice as far at roughly $4{,}000\ \text{km/s}$. With $H_0 \approx 70\ \text{km/s/Mpc}$, a galaxy at $50\ \text{Mpc}$ recedes at about $3{,}500\ \text{km/s}$, giving a redshift $z \approx v/c \approx 0.012$, so a spectral line emitted at $500\ \text{nm}$ arrives near $506\ \text{nm}$.
Key Insight
Galaxies are not flying through space away from us; space itself is stretching and carrying them along, and every galaxy sees the same pattern of everything else receding, so no galaxy is the center. Expansion does not stretch gravitationally bound systems. The Milky Way, the solar system, and atoms are held together by forces that overwhelm the expansion, and only the gaps between galaxy clusters grow.