Balance shafts are most common in inline four cylinder (straight-4) engines which, due to the asymmetry of their design, have an inherent second order vibration (vibrating at twice the engine RPM) which, contrary to popular belief, cannot be eliminated no matter how well the internal components are balanced. This vibration is generated because the movement of the connecting rods in an inline engine is not symmetrical throughout the crankshaft rotation; thus during a given period of crankshaft rotation, the descending pistons and ascending pistons are not always completely opposed in their acceleration, giving rise to a net vertical inertial force twice in each revolution whose intensity increases exponentially with RPM, no matter how closely the components are matched for weight. See the 0 degree cylinder angle, 180 degree crankshaft angle animated example here for a very clear depiction of this sometimes hard to visualize vibration (as well as the mathematical equation which describes it).

The problem increases with larger engine displacement, since the only ways to achieve larger displacement are with a longer piston stroke, increasing the difference in acceleration, or by a larger bore, increasing the mass of the pistons; either way, the magnitude of the inertial vibration increases. For many years, two litres was viewed as the 'unofficial' displacement limit for a production inline four cylinder engine with acceptable NVH characteristics. The development of the General Motors 2.3 litre Quad 4 engine in 1987, described as "rough as a cob" by one automotive reviewer, and its subsequent development into the much more positively received 2.4 litre version with balance shafts confirms the wisdom of this assessment.

The basic concept behind balance shafts has existed for nearly a century and is no longer patentable. Two balance shafts rotate in opposite directions at twice engine speed. Equally sized eccentric weights on these shafts are sized and phased so that the inertial reaction to their counter-rotation cancels out in the horizontal plane, but adds in the vertical plane, giving a net force equal to but 180 degrees out of phase with the undesired second-order vibration of the basic engine, thereby canceling it. (Some motorcycle enthusiasts believe that Honda's original application of this technology to their V-twin motorcycle engine overly damped out the vibration, giving an excessively 'dead' feel, so that they later reduced the size of the balance shafts in order to furnish the rider with some feedback as to engine speed).

The actual implementation of the concept, however, is concrete enough to be patented. The basic problem presented by the concept is adequately supporting and lubricating a part rotating at twice engine speed, at the higher RPMs where the second order vibration becomes unacceptable. Mitsubishi Motors pioneered the design in the modern era with its "Silent Shaft" Astron engines in 1975, with balance shafts located low on the side of the engine block, driven by chains from the oil pump, and subsequently licensed the patent to Porsche, then to other manufacturers. Since then, other manufacturers have adapted the same basic layout to their needs.
Saab has further refined the balance shaft principle to overcome second harmonic sideways vibrations (due to the same basic asymmetry in engine design, but much smaller in magnitude) by locating the balance shafts with lateral symmetry but at different heights above the crankshaft, thereby introducing a torque which counteracts the sideways vibrations at double engine RPM, resulting in an exceptionally smooth four cylinder engine.
There is some debate as to how much power the twin balance shafts cost the engine. The basic figure given is usually around 15 horsepower (11 kW), but this seems excessive for pure friction losses. It is likely that this is a miscalculation derived from the common use of an inertial dynamometer, which calculates power from angular acceleration rather than actual measurement of steady state torque. The 15 horsepower (11 kW), then, includes both the actual frictional loss as well as the increase in angular inertia of the rapidly rotating shafts, which would not be a factor at steady speed. Nevertheless, many owners modify their engines by removing the balance shafts, both to reclaim some of this power, but also to reduce complexity and potential areas of breakage for high performance and racing use. As mentioned above, it is commonly believed that the smoothness provided by the balance shafts can be attained after their removal by careful balancing of the reciprocating components of the engine, but that stems from a basic misunderstanding of their operation.

Will I feel extra vibration once I have eliminated my balance shafts? 
Eliminating the balance shafts will not be for everyone. However the vibration from eliminating the balance shafts is unnoticeable to some people and is less than the increased vibration you feel from poly motor mount inserts or mounts. In over 3 years of eliminating balance shafts people are continually surprised that the vibration is minimal.