Usually, when we try to understand an object, we like to divide the object into smaller pieces (or more basic components). If we can do that, we say that we gain a better understanding of the object. This is the reductionist approach.

But there is another way of understanding. We do not think about the internal structure of the object, and pretend the internal structure is not there. (Maybe the internal structure really does not exist.) We try to understand an object through all its relations with all other objects.

In fact, we use all those relation to define the object. In other words, there are no objects, just relations. A collection of relations defines the notion (and all the properties) of an object.

That's it. What we just described is the category theory.

One physics example: How do we understand different phases of matter? A categorical way is to understand the phase transitions between different phases, and use the phase transition to define different phases.
The second physics example: Using wavefunction to understand our quantum world is a reductionist approach. In a categorical approach, we want to use all the physical measurements, and only measurements (measurements = correlation functions = relations), to describe quantum theory. So to categorify quantum mechanics is to abandon wavefunctions, and to only use measurable quantities to formulate the quantum mechanics.

吴咏时老师的社会学例子：范畴学的精神像马克思说过的，人这个个体是通过人和人的关系定义的。-- 来自  根源之渦

One application of the category theory is to describe the unification of matter, information, and geometry: matter <--> information --> geometry  (see 物理中的近代数学 Modern mathematics in physics and 我们生活在一个量子计算机里 We live inside a qubit world )  [Note that a category is a collection of arrows (the relations) that connect objects. So matter <--> information --> geometry is a category. ]

Category theory represents a way to understand our world via relations. Category theory is a theory about the relations. 范畴学是关系学.

(For a mathematical point of view, see Why do we need category theory?  )

(A real mathematical definition of category.)

A story:  My own experience of learning category theory

During our work on string-net condensation in 2004, we realized that string-net condensation and tensor category are  closely related. So I tried to learn tensor category from a math textbook. After reading a few sentences, I saw a word "morphism" which I did not know what does it mean. So I skipped it. But a few sentences later,  "morphism" appeared again and I skipped it again. Then in the next sentence, I was hit by "morphism" the third time. I gave up, before going through half page of the textbook.

Later, I attended many lectures/talks by my mathematician friends, about the arrows and points. I felt I understood something during the lectures, but a few hours later, I forgot everything, and I still did not get it. But after each lecture, I gained a little more understanding. It took me 10 years to get to the level of understanding represented by this article, which is pretty simple. I wonder why it took me so long?!