While the concept of galaxies as "island universes" can be traced back to Wright (1750) and Kant (1775) the study of the formation of galaxies did not begin until after their extra-Galactic status was confirmed by Hubble (1929). In fact, much of the early work on galaxy evolution and formation was driven by the necessity of understanding galaxies in order to answer questions of cosmology (such as whether or not the Universe began with a Big Bang). While an understanding of galaxies remains necessary for such reasons even today, the field has since become an important one in its own right.
Modern galaxy formation theory therefore grew out of early studies of cosmology and structure formation and is set within the cold dark matter cosmological model and so proceeds via a fundamentally hierarchical paradigm. Observational evidence and theoretical expectations indicate that galaxy formation is an ongoing process which has been occurring over the vast majority of the Universe's history. The goal of galaxy formation theory then is to describe how simple physics gives rise to the complicated set of phenomena which galaxies encompass.
Galaxy formation is very much an observationally driven field in the sense that we are still decidedly in the stage of making new experimental discoveries rather than performing precision tests of well-specified theoretical models. While this situation shows signs of a gradual shift to the "precision tests" phase it seems unlikely that the transition will be completed any time soon. In addition, astronomy is perhaps uniquely hindered by experimental biases, since we are not able to design the experiment, merely to observe what the Universe has decided to put on show. The complicated nature of the resulting selection effects result in a secondary, but very important, role for theoretical models, namely in quantifying these biases and interpreting the data. While this secondary role is well established it needs to become more so, in particular it should become an integral part of any observational campaign and will require direct and simple access to modeling capabilities for all.