The main difference between the lagrangian and eulerian descriptions is that in the lagrangian description, the motion of each fluid particle is tracked to observe changes in its properties, and in eulerian consideration, the changes in properties of the fluid are observed at a fixed point in space.
Lagrangian method:
In the lagrangian method, it is considered that the fluid is consist of a number of fluid particles, and each particle or fluid parcel is tracked throughout its motion to measure changes in its properties with a variation of time.
Consider an observer sitting in a car, and observing the displacement, velocity, and acceleration of the car throughout the time interval. This is an example of the lagrangian approach.
A lagrangian description is also known as material description as it is following a material (fluid) particle. In this method, the fluid motion is described by the properties as a function of time.
∴ V = V(t)
Eulerian method:
In eulerian consideration, the measurements are carried at a certain fixed point in space and the changes in properties are measured with the variation of time. The eulerian consideration or description is also known as field description.
The eulerian consideration never focuses on particular fluid parcels whereas it observes the properties of the fluid passing from a particular fixed point.
Consider you are sitting near the window of your classroom and watching the cars outside of the window, this is an example of the eulerian description.
Therefore in eulerian consideration, the fluid properties are the function of space and time.
∴ V = V(x, y, z, t)
The below figure shows the measurement of a fluid velocity by using a pitot tube fixed at a certain point. This is an example of the eulerian approach.
Lagrangian vs Eulerian:
Sr. No.  Lagrangian method  Eulerian method 

1  In Lagrangian, the observer follows a fluid particle/parcel to observe changes in its properties.  In the eulerian approach, the fluid is observed through a fixed point or control volume. 
2  The fluid properties are observed at various locations in space wherever the fluid parcel moves.  The changes in properties of the fluid are measured at a fixed point. 
3  The fluid properties are the function of the initial position of the particle and time.  The fluid properties are the function of space and time. 
4  It is a control mass approach.  It is a control volume approach. 
5  In the Lagrangian approach, measurements are difficult as it focuses on the movement of each fluid particle and the interaction between them.  The eulerian demonstration is easy in comparison with the lagrangian. 
6  Lagrangian consideration has fewer applications.  The eulerian consideration is used in many practical applications. 
FAQs:

Which approach is most commonly used? lagrangian or eulerian?
The eulerian approach is most commonly used than the lagrangian approach.

How fluid properties are different in the lagrangian and eulerian approaches?
In the lagrangian approach, the fluid properties are the function of time while in the eulerian approach the properties are the function of time and space.