Mode shape is the the shape or the displacement pattern of the element or a structural systems it attains when it vibrate at its corresponding natural frequency.
The mode shape is also known mathematically known as eigen vector and its corresponding frequency is known as eigen frequency.
The eigen vector and the eigen frequency is solution of any eigen value problem [A]{X}=lambda{X} .
This eigen value problem or the eigen equation is obtained in several engineering problem and in structural engineering it is most commonly found in the dynamic equation and the buckling equation.
For the dynamic equation it is simply expressed as [K-(ω^2)M]{x}=0 where K is the stiffness matrix, M is the mass matrix and ω is the natural frequency.
If there are N number of degree of freedom in a structure then there would be n number of mode shapes and their corresponding natural frequencies.
The combination or the superposition of all the mode shapes at a particular instant of time would give you the idea of the resultant displaced pattern of the entire structure.
As discussed above, that the mode shape is the response when a structure is perturbed . So such type of perturbation can be of different types like ground excitation by earthquake action, harmonic loading imposed by any rotating machine sitting on a structure or any blast loading.
So when ever the structure is perturbed, one need to first know what would be its overall response and for the very same reason one has to determine all the different mode shapes which shape to mathematically include them to form the modal matrix which is required un-coupling the multiple number of coupled dynamic equation
[M]{x"} + [C]{x'} + [K]{x} = - {P(t)}
Where [M]is the mass matrix [ C] is the damping matrix , [K is the stiffness matrix and - {P(t)} is the external dynamic load which varies with time.
In general if you are using the Response spectrum analysis then you need to refer to the mass participation factor and most of the design codes suggests to consider the total contribution of percentage of mass from all the selected modes should be at-least 90% whereas if you are using the Time- History analysis, the concern is to refer the Modal participation factor (insert SET PART FACT in the input editor before the JOINT COORDINATES).So, in general regarding the the mode shapes its your engineering judgement.
For more information you may refer to the standard textbook on dynamic analysis by the the author Mario Paz, A.K chopra or Clough and Penzein.