Black holes have so much gravity that even light can't escape from them. If we can't see them, and the suck up all electromagnetic radiation, then how can we find them?
To add to John Conde's answer. According to the NASA web page "Black Holes", detection of black holes can obviously not be performed by detection any form of electromagnetic radiation coming directly from it (hence, can not be 'seen').
The black hole is inferred by observing the interaction with surrounding matter, from the webpage:
This also includes detection of x-ray radiation that radiates from matter accelerating towards the black hole. Although this seems contradictory to my first paragraph - it needs to be noted that this is not directly from the black hole, rather from the interaction with matter accelerating towards it.
One way is by following Gamma Ray Bursts. When a black hole feeds on surrounding gas or swallows a star that got too close, they often emit gamma ray bursts which are very energetic and easy to spot (although they don't last long).
In the case of super massive blackholes, they are seemingly at the center of every medium and large galaxy. It makes where to look rather easy.
A black hole can also be detected by how it bends light as various bodies move behind it. This phenomenon is called gravitational lensing, and is the most visually stunning prediction of Einstein's theory of General Relativity.
This image portrays the geometry of gravitational lensing. Light from luminous background objects are bent due to the warping of space-time in the presence of mass (here, the red dot could conceivably be the black hole in question):
Astronomers have discovered the existence of a super-massive black hole at the center of our very own Milky Way Galaxy, and has been dubbed Sagittarius A*.
Over a period of ten years, the trajectories of a small group of stars have been tracked, and the only explanation for their rapid movement is the existence of a highly compact object with the mass of about 4 million suns. Given the mass and distance scales involved, the conclusion is that it must be a black hole.
There are many, many ways of doing this.
This is by far the most well known. It has been mentioned by the others, but I'll touch on it.
Light coming from distant bodies can be bent by gravity, creating a lens-like effect. This can lead to multiple or distorted images of the object (Multiple images give rise to Einstein rings and crosses).
So, if we observe a lensing effect in a region where there isn't any visible massive body, there's probably a black hole there. The alternative is that we are peering through the dark matter 'halo' which surrounds (and extends passed) the luminous components of every galaxy and galaxy cluster (See: Bullet Cluster). On small enough scales (i.e. - the central regions of galaxies), this is not really an issue.
(This is an artist's impression of a galaxy passing behind a BH)
Spinning black holes and other dynamical systems involving black holes emit gravitational waves. Projects like LIGO (and eventually, LISA) are able to detect these waves. One major candidate of interest for LIGO/VIRGO/LISA is the eventual collision of a binary black hole system.
Sometimes we have a black hole in a binary system with a star. In such a case, the star will orbit the common barycenter.
If we observe the star carefully, its light will be redshifted when it is moving away from us, and blueshifted when it is coming towards us. The variation in redshift suggests rotation, and in the absence of a visible second body, we can usually conclude that there's a black hole or neutron star there.
Salpeter-Zel'dovitch / Zel'dovitch-Novikov proposals
Going in to a bit of history here, Salpeter and Zel'dovitch independently proposed that we can identify black holes from shock waves in gas clouds. If a black hole passes a gas cloud, the gases in the cloud will be forced to accelerate. This will emit radiation (X-rays, mostly), which we can measure.
An improvement on this is the Zel'dovitch-Novikov proposal, which looks at black holes in a binary system with a star. Part of the solar winds from the star will be sucked in to the black hole. This abnormal acceleration of the winds will, again, lead to X-ray shock waves.
This method (more or less) led to the discovery of Cyg X-1
Cyg A is an example of this. Spinning black holes act like cosmic gyroscopes — they do not easily change their orientation.
In the following radio image of Cyg A, we see these faint gas jets emanating from the central spot:
These jets are hundreds of thousands of light years long — yet they are very straight. Discontinuous, but straight. Whatever object lies at the center, it must be able to maintain its orientation for very long.
That object is a spinning black hole.
Most quasars are thought to be powered by black holes. Many (if not all) of the candidate explanations for their behavior involve black holes with accretion disks, e.g. the Blandford-Znajek process.
All 4 answers given prior to this one are very good and complete each other; finding an object orbiting your target object enables you to also calculate the mass of your target object.
Matter falling into a black hole is accelerated toward light speed. As it is accelerated, the matter breaks down into subatomic particles and hard radiation, that is, X-rays and gamma rays. A black hole itself is not visible, but the light (mostly X-rays, gamma rays) from infalling matter that is accelerated and broken up into particles is visible.
By looking toward the center of our galaxy, the Chandra X-ray space telescope has observed several black holes besides Sgr A*, indirectly, by catching the hard radiation of infalling matter flaring up as they swallow something; afterward, the black holes go dark again if there is nothing more to assimilate nearby;
Here you can see some of this flaring in the swarm of black holes near the center of our galaxy.
Methods to detect black holes (which are not really holes or singularities, as they do have mass, radius, rotation, charge and hence density, which varies with radius, see http://en.wikipedia.org/wiki/Schwarzschild_radius ).
But there will be nothing left to see what caused it; better have some observations of that spot in the sky before it happens.
Another thing to point out is that these observational methods typically set a density limit on the invisible object such that the only thing that would reasonably explain it would be a black hole.