While stellar parallax can only be used to measure distances to stars within hundreds of parsecs, Cepheid variable stars and supernovae can be used to measure larger distances such as the distances between galaxies.
This video, Measuring the Universe, gives a great introduction to this topic.
Measuring the Universe from Royal Observatory Greenwich on Vimeo.
Using Cepheid Variables to Measure Distance
Cepheid variable stars are intrinsic variables which pulsate in a predicatable way. In addition, a Cepheid star's period (how often it pulsates) is directly related to its luminosity or brightness.
Cepheid variables are extremely luminous and very distant ones can be observed and measured. Once the period of a distant Cepheid has been measured, its luminosity can be determined from the known behavior of Cepheid variables. Then its absolute magnitude and apparent magnitude can be related by the distance modulus equation, and its distance can be determined. Cepheid variables can be used to measure distances from about 1kpc to 50 Mpc.
For example, if an astronomer observed a Cepheid star with period of 34 days, comparing to previously measured Cepheids, its absolute magnitude is -5.65. If its apparent magnitude was +23.0, the astronomer could use the distance modulus equation:
m - M = 5 log d - 5
d = 10(m-M+5)/5 parsecs
to find the distance to the Cepheid:
|d = 10(23-(-5.65)+5)/5 parsecs
|d = 5.4 × 106 parsecs
Using Type Ia Supernovae to Measure Distance
Type Ia supernovae are all caused by exploding white dwarfs which have companion stars. They can be distinguished from other supernovae because they do not have hydrogen lines in their spectra and have a strong Si II line at 615 nm. The gravitational pull of the white dwarf causes it to take matter from its companion star. Eventually it reaches a high enough mass (about 1.44 solar masses) that it cannot support itself against gravitational collapse and explodes. All type Ia supernovae reach nearly the same brightness at the peak of their outburst with an absolute magnitude of -19.3±0.03. They then follow a distinct curve as they decrease in brightness. So when astronomers observe a type Ia supernova, they can measure its apparent magnitude, knowing what its absolute magnitude is. They can then use the distance modulus to calculate the distance to the supernova, and the galaxy that it is in. Type Ia supernovae can be used to measure distances from about 1 Mpc to over 1000 Mpc.
An example to try
A Cepheid variable star has a period of 3.7 days, and from this we know its absolute magnitude is -3.1. Its apparent magnitude is 5.5. How far away is this Cepheid variable star?
This star is 524.8 parsecs away.