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Star formation is the process by which hydrogen & helium in molecular clouds change into the ball of plasma we call a star.

According to current theories of star formation, cores of molecular clouds (regions of especially high density) become gravitationally unstable, fragment and begin to collapse (sometimes, shockwaves from supernovae will trigger star formation in nearby gas clouds). Part of the gravitational energy lost in this collapse is radiated in the infrared, with the remainder increasing the temperature of the core of the object. The accretion of material happens partially through a circumstellar disc. When the density and temperature are high enough, deuterium fusion ignition occurs, and the outward pressure of the resultant radiation slows (but does not stop) the collapse. Material comprising the cloud continues to "rain" onto the protostar. In this stage bipolar flows are produced, probably an effect of the angular momentum of the infalling material. Finally, hydrogen begins to fuse in the core of the star, and the rest of the enveloping material is cleared away.

The stages of the process are well defined in stars with masses around one solar mass or less. In high mass stars, the length of the star formation process is comparable to the other timescales of their evolution, much shorter, and the process is not so well defined. The later evolution of stars are studied in stellar evolution.

Observations
Key elements of star formation are only available by observing in wavelengths other than the optical. The structure of the molecular cloud and the effects of the protostar are best observed in rotational transitions of CO and other molecules; these are observed in the millimeter and submillimeter range. The radiation from the protostar and early star has to be observed in infrared astronomy wavelengths, the extinction caused by the rest of the cloud where it is being formed is usually too big to allow us to observe it in the visual part of the spectrum.

The formation of individual stars can only be directly observed in our Galaxy, but in distant galaxies star formation has been detected through its unique spectral signature.

References & Further reading
The theory of collapse of gas under its own gravity was developed by Jeans and although the theory does not treat many phenomena known to be important it is a very useful and widely used first approximation. The original paper is available free of charge (to participating institutions):

J. H. Jeans. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 199. (1902), pp. 1-53.
The theory of solar mass star formation is detailed in:

Shu, Frank H.; Adams, Fred C.; Lizano, Susana. Star formation in molecular clouds - Observation and theory. Annual review of astronomy and astrophysics. Volume 25 (A88-13240 03-90). Palo Alto, CA, Annual Reviews, Inc., 1987, p. 23-81.