There are many independent lines of evidence that most of the matter
in the universe is dark.  Essentially, many of these measurements rely
on "weighing" an object such as a galaxy or a cluster of galaxies by
observing the motions of objects within it, and calculating how much
gravity is required to prevent it flying apart.

(1) Rotation patterns in spiral galaxies.
(2) Velocities of galaxies in clusters.
(3) Gravitational lensing.
(4) Hot gas in galaxies and clusters.
(5) Large-scale motions. 

(1) Rotation patterns in spiral galaxies. The disks of spirals are
full of stars and gas in nearly circular coplanar orbits, making them
wonderful tracers for the gravitational field in which they move.  In
centrally-concentrated masses, such as within the solar system (where
most of the mass is concentrated in the Sun), the
velocity-vs.-distance relation approaches Kepler's 3rd Law, velocity^2
= constant * central mass / distance.  Once we sample outside the
central concentration of stars, using observations of the 21cm line
emitted by neutral hydrogen clouds, spiral galaxies violate this
velocity-distance relation quite flagrantly; velocity=constant is a
good approximation (hence the moniker "flat rotation curves").  A
sample picture and rotation curve is at
<URL:http://crux.astr.ua.edu/gifimages/ngc5746.html>. To get this
pattern, one needs a mass distribution that goes as density
proportional to 1/radius^2, much fluffier than the observable stars
and gas in the galaxy, and in an amount that may be 10 or more times
the total mass we can account for with stars, dead stellar remnants,
gas, and dust.  There were hints of this issue for a while, but it was
a series of observations by Vera Rubin and collaborators in the
mid-1970's that really rubbed our noses in it.

(2) Velocities of galaxies in clusters.  Galaxies in clusters have
random orbits.  By measuring the dispersion for, e.g., 100 galaxies in
the cluster, one finds typical dispersions of 1000 km/s. The clusters
must be held together by gravity, otherwise the galaxies would escape
in less than 1 billion years; cluster masses are required to be at
least 10 times what the galaxies' stars can account for.  This problem
was first demonstrated in 1938 by Fritz Zwicky who studied the
galaxy-rich Coma cluster.  Zwicky was very bright, very arrogant, and
highly insulting to anyone he felt was beneath him, so this took a
long while to sink in. Today we know that virtually all clusters of
galaxies show the same thing.

(3) Gravitational lensing. General relativity shows that we can treat
gravity (more precisely than in Newtonian dynamics) by considering it
as a matter-induced warping of otherwise flat spacetime. One of the
consequences of this is that, viewed from a distance, a large enough
mass will bend the paths of light rays.  Thus, background objects seen
past a large mass (galaxy or cluster of galaxies) are either multiply
imaged or distorted into "arcs" and "arclets."  Some beautiful
examples can be seen at
<URL:http://www.stsci.edu/pubinfo/PR/96/10/A.html>,
<URL:http://www.stsci.edu/pubinfo/PR/95/14.html>, and
<URL:http://www.stsci.edu/pubinfo/PR/95/43.html>.  When we know the
distances of foreground and background objects, the mass inside the
lensing region can be derived (and for some of these multi-lens
clusters, its radial distribution). Same old story - we need a lot
more mass in invisible than visible form.

(4) Hot gas in galaxies and clusters. A real shocker once X-ray
astronomy became technologically possible was the finding that
clusters of galaxies are intense X-ray sources. The X-rays don't come
from the galaxies themselves, but from hot, rarefied gas at typically
10,000,000 K between the galaxies.  To hold this stuff together
against its own thermal motions requires - you guessed it, huge
amounts of unseen material.

It is worth noting that these last three methods all give about the
same estimate for the amount of dark matter in clusters
of galaxies.  

(5) Less direct evidence also exists: On larger scales, there is
evidence for large-scale "bulk motions" of galaxies towards
superclusters of galaxies, e.g., the Great Attractor.  There is also
the question of reconciling the very small (1 part in 100,000)
observed fluctuations in the cosmic microwave background with the
"lumpy" galaxy distribution seen at the present day; dark matter helps
nicely to match these two facts because the density fluctuations grow
more rapidly with time in a higher-density Universe.  Finally, the
theory of inflation (which is an "optional extra" to the standard big
bang model) usually predicts that the universe should have exactly the
critical density, which could require as much as 95% of the mass in
the Universe to be dark.

It is worth mentioning the possibility of non-standard gravity
theories, which attempt to explain the above list of observations
without dark matter. It turns out that modifying the inverse-square
law of gravity does not work well, essentially because the dark matter
problem extends over so many different lengthscales. Modifying the F =
ma law has been tried, e.g., by Milgrom, but relativistic versions of
this theory have not been found, and most cosmologists are reluctant
to abandon Einstein's GR which is elegant and well tested (at least on
solar system scales).