Why are galactic centers always brighter than the edges? As you can see the image below and other galaxy images, the center is generally much brighter. 

Why is that? 
Is there a very big star? A very big gravitational field?
 A: Simply, the galaxies are thicker in the middle (that is, they bulge further above and below the ecliptic) and have a higher density of stars and gas towards the core.
There are exceptions to the 'brighter in the center' observation, such as this beauty from the Arp 147 pair imaged by the Chandra observatory, which is the result of a galactic collision:

A: Both elliptical and spiral galaxies have a stellar mass profile which have a well-defined maximum. In addition, both types of galaxy have symmetries - axial in the case of spirals and either axial or triaxial in the case of ellipticals. Since, in general terms, the luminosity tends to follow the stellar mass, this means that their projected brightness on the plane of the sky is centrally concentrated after integrating over the hidden dimension. In addition spiral galaxies often have a nuclear "bulge" that enhances this effect. There is also the issue of active galactic nuclei. The supermassive black hole that is present at the centres of many, if not most, galaxies will emit copious light if it is being "fed". In many instances, the AGN luminosity dominates the overall light from a galaxy, inevitably leading to a very bright central region. For a long time it was unclear whether "quasars" were in fact point-like, but we now know they are at the centres of galaxies.
There are of course exceptions and complications. Colliding galaxies, discussed by Richard Terrett, are one example. Their distorted morphologies are because they are in the midst of an interaction and their potentials have lost their neat symmetries. However, there are other more common examples, were things are also not so neat.
The idea that mass follows light, or rather vice-versa is only an approximation. High mass stars have a much smaller mass-to-light ratio, but high-mass stars are short lived and so are seen more-or-less where they are born. They do not have time to "feel" the galactic potential and adopt any particular symmetry. Therefore if there is vigorous star formation, the galactic luminosity can be dominated by pockets or rings or bars of high-mass star formation. Such "starburst galaxies" will also have lots of (possibly asymmetric) dust that can obscure the galactic plane or centre. A famous example - M82 is shown below (from http://www.calvin.edu/academic/phys/observatory/images/Astr212.Spring2007/ ).

A further contradiction are the Irregular galaxies. The best known examples are the Large and Small Magellanic clouds -though the classification is still in a state of flux. What is not argued is that these galaxies do not conform to the usual Hubble classification types and are usually not brightest in the centre. An example is IC559 (as seen by HST) an irregular galaxy with some signs of spiral structure.

