Black holes are a finicky lot. They are thought to exist at the center of most active galaxies and are predicted to exist by general relativity. What we theorize black holes to be are supermassive objects "power[ing] galactic nuclei by converting the gravitational energy of accreting matter into radiation" (http://www.nature.com/nature/journal/v455/n7209/full/nature07245.html). But, there's a problem currently with proving the existence of black holes - we cannot see them. We have seen the effects of intense gravity (See the article about Stellar Motion around the black hole located at Sagittarius A* at our galaxy's center.), but nothing definitive yet as to what exactly is the cause. Theoretically it is possible to observe the presence of a black hole by a process known as "gravitational lensing," an effect predicted by general relativity, whereby light from some source travels from the source and is bent by the extreme gravity of an object, such as a black hole or a galaxy. In the case of a black hole, as the black hole comes between some distant bright object and an observer the image of the object will grow in size, making it appear larger, and brighter due to the change in angular size while in line.

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However, there is a tremendous amount of electromagnetic radiation emitted from the activity near black holes, and so the most promising possibilities in observing black holes are in the range of x-ray and radio frequencies. What we are actually trying to observe is a relatively small silhouette amongst massive amounts of bright, hot matter. Quasars, which are massive conglomerations of bright, hot matter located at the center of some very distant galaxies, are currently thought to be caused by the gravity of black holes at the center, feeding off an accretion disk of matter and subsequently converting that energy to radiation. Inconveniently for us, this incredible amount of matter near a black hole also serves the purpose of clouding most any imaging of the black hole silhouette.
X-ray telescopes and radio telescopes are our best bet currently for observing black holes. The most recent efforts in imaging black holes have been attempts to image Sagittarius A*, the supermassive black hole thought to occupy the center of the Milky Way.