Eulerian Description and Lagrangian Description
Definition 1
In particular, in fluid mechanics, the fluid has no definite shape and its state of motion is difficult to know in detail, so one postulates something like a fluid particle. There are two descriptions for describing the state of motion of a fluid particle.
Eulerian description
$$ {\frac{ \partial u }{ \partial t }} $$
The Eulerian description fixes a moment in time and observes the state of motion of the fluid at every point.
Lagrangian description
$$ {\frac{ \partial u }{ \partial x }} , {\frac{ \partial u }{ \partial y }} , {\frac{ \partial u }{ \partial z }} $$
The Lagrangian description follows individual fluid particles and observes their states of motion.
Explanation 2
The term description may also be called motion or specification, but the exact word is not very important; what matters is the context that there are two ways to describe the motion from the fluid’s point of view.
The Eulerian and Lagrangian descriptions are not opposing but complementary concepts, and it is important to understand that both perspectives are needed in fluid mechanics, where partial differential equations are indispensable.
Fundamentally, the Eulerian description—which stops time and contemplates the phenomenon—is more intuitive, but whenever any physical quantity interacts spatially (in any direction), an idea like the Lagrangian description is necessary. For example, imagine a junction where two pipes with different flow speeds merge into one: with only the Eulerian description, one can imagine a situation in which it is impossible to know the speeds at which the fluid is entering from each pipe.