The subject of the sentence is what the sentence is all about or what the sentence is talking about.

For example: The dog is running down the long hallway.

*The sentence is talking about a DOG.

An object of a sentence could be:

The man played with the dog before he was given away.

DOG is the object because the sentence is not about the dog. DOG is an object that has to do with the MAN (which is the subject).

The MAN played with the DOG. Therefore, the man is the subject and dog is the object.

*TIP: Most of the time the subject comes before the object. However there are some that contrast this observation:

Example: In the house, the man played his piano. House is an object, while man is the subject.

TIP: A verb usually comes after the subject.

ex. Hilary put her backpack on the table.

The subject is Hilary because she is the dominant subject in the sentence. The sentence focuses on her and the table is merely just extra detail.

I just found something on the internet:

The Subject of a sentence is what the sentence is about and for that reason is usually the first noun or noun phrase in the sentence.

ex. “I like Fernando.”

The Object form is used after the verb but also after prepositions: “Fernando likes me” or “Fernando likes working with him.”

Sorry if it’s a bit confusing. I listed the source if you want to check out the website. It goes into more detail.