As we have described a black hole, we never can observe one of them because they would not reflect or emit any radiation or particle. But there are certain effects that can be detected. One of these effects is the gravitational effect on a neighboring star.
Suppose a binary star system (two stars so close to turning the one around the other) in which one of the stars is visible and than we can calculate its distance from Earth, and its mass. The visible star will make a few oscillatory movements in space due to the gravitational attraction of the invisible star. From these movements we can calculate the mass of the invisible star .
If this invisible star exceeds a mass of about 2.5 times the mass of our Sun, we must assume that it’s a black hole.
In addition if the visible star is close enough, it could give him 
part of its mass that would fall into the black hole being accelerated at such speed that it would reach a very high temperature such to emit x-rays. But this would also happens if it were a neutron star instead of a black hole.
An example of a detected object that meets the two first conditions exposed is the binary star system called Cignus – X 1, which is a source of very intense x-ray formed by a visible star and another invisible star with a calculated mass exceeding 2.5 solar masses. It has been also detected objects of thousands of solar masses in the centers of galaxies, supermassive black hole candidates.
Space telescope Hubble showed us this image of the Galaxy M87 where we can see a huge jet of gas allegedly issued by a central super-massive hole accretion disk
Apart from this it must also take into account that S. Hawking deduced that a black hole would produce subatomic particles in their vicinity, losing mass and radiating such particles, which would be another way of detection.
We read in black holes and small universes Stephen Hawking, in his lecture “The future of the universe” saying:
“The principle of quantum mechanics indetermination indicates that particles cannot have simultaneously very defined position and speed. The greater the accuracy with which define the position of a particle, the lower the accuracy with which determines its speed and vice versa. If a particle is in a black hole, its position is very defined there, meaning that its speed cannot be exactly defined. It is possible that the speed of the particle is higher than light and in this way could escape from the black hole.”
But we must not think that the hole would lose mass, because a black hole of a few solar masses would emit lower radiation than the cosmic background of the universe, receiving more energy that it would issue, and therefore increasing their mass eventually.
In addition to observation of the movement of the stars to detect neighboring invisible stars with big mass that may be black holes, or by the radiation emitted by the accretion disks, we can also have tracks of black holes by the gravitational lens effect, because a black hole would divert the light of a nebula that is found behind, as it would occur quite visible figures in the form of arc or circle.
Simulation of how a galaxy behind a black hole would be.
[Source: in Spanish http://www.relatividad.org/bhole/deteccion.html]