We recently compiled all of our vibration analysis seminars and white papers from over the years into one comprehensive document on vibration analysis. It's posted at AppliedCAx.com, where one can find all of our FEMAP resources. Vibration analysis is a huge topic and is easily the second most common type of FEA analysis after basic static stress analysis.
Within the field of vibration analysis, the most common type of analysis is that based on the linear behavior of the structure or system during its operation. That is, its stress/strain response is linear and when a load is removed, the structure returns to its original position in a stress/strain free condition.
Although this might sound a bit restrictive, it actually covers a huge swath of structures from automobiles, planes, ships, satellites, electrical circuit boards and consumer goods. If one needs to consider a nonlinear response of the structure during operation, there exist codes such as LS-DYNA that can solve for the complete nonlinear vibration response. But that is not simple or basic and is left for another seminar sometime in the future.