You know what I mean by the grammatical subject. "The subject of the sentence." Oh, yeah, no problem. It's the agent, the do-er, the entity or person or thing that engages in whatever the verb says. It's always a noun phrase. Examples:
He hit me.
Reyes tried to escape.
The white-furred cat jumped over the fence.
If you only consider it from the perspective of its grammatical definition, though, you might miss its most important function. It focuses reader attention and gives special importance to whatever magical noun phrase gets that all-important, first-in-the-sentence spot*. When we choose to make something the subject of a sentence, we're exercising a great power. *(I'll consider exceptions below)
I'm deliberately going to quote Spiderman: "With great power comes great responsibility."
The easiest way to see the power of the subject demonstrated is by looking at what happens when we use it in unfortunate ways.
Teleporting readers into the air
Readers will be looking to ground themselves at the beginning of any story. What you chose to use as the subject of your first sentence thus becomes very important. If you begin, "The apartments at 200 Smith Street," then your readers will find themselves floating over said apartments. If, on the other hand, you begin, "I couldn't believe my eyes," then your readers will find themselves looking through the eyes of a person, "I," about whom they'll be looking to learn more. If you give them a name, like "George found the body at midnight," then they'll instantly be transported to George's location (beside him, or in his head, yet to be determined). An enormous amount about your narrator will be evident very quickly. Because I do very close internal point of view, I'm always tempted to start with internalization in my first sentence (implying the presence of a character rather than showing it). However, I always try to get the name of my protagonist in as the subject of a sentence within the first paragraph - and usually in the first sentence. Since I don't intend my readers to float on air, I don't want to transport them there accidentally.
Losing readers in a trance
The subject of the sentence can be a noun phrase, and it can vary in length, but because we're snagging a reader's close attention with it, if we let it get too long we've got trouble. Check out the difference between these two sentences.
The white-furred cat jumped over the fence.
The white-furred cat which my brother found over Christmas break and nursed back to health with the help of three friends jumped over the fence.
I often try to add extra information into the background of a sentence by using long noun phrases, but there's a limit to how much you can do without having your reader hit the word "jumped" above and go, "What?" They're engaged in trying to figure out precisely whom they'll be watching for the next few sentences (as subjects usually establish referents that get carried forward) and will follow the details... and when the verb finally breaks them out of the trance, they may no longer have any idea where they are or what you were saying! It's good to watch out for that.
Telekinetically striking readers over the head
A grammatical subject is a strong statement. By placing someone in grammatical subject position at the start of a paragraph, you're essentially saying, "Reader, you'll be hearing about this person for the next few sentences." This means you don't need to do it more than once. I talked some time ago about the hierarchy of reference. The hierarchy of reference basically says that you use a name for someone the first time you mention them, and then typically a pronoun thereafter unless you have to disambiguate between several possible pronouns, in which case you can use a brief description (more extensive details are here). A possible sequence of subjects might therefore be: Tagret, he, he, the noble boy. Now, imagine what would happen if you said, "Tagret, Tagret, Tagret, Tagret." By the end of it your reader would be begging for mercy. The same effect can also be achieved (far more easily) with the pronoun I: "I, I, I, I, I, I." Ay-ay-ay! Have mercy on your reader and don't always use the identical subject, but vary your sentences.
Transforming readers into fish
Think about the close attention that the subject demands, and then ask yourself where you're putting it. If your reader is working through the first paragraph of a story, and the first two or three sentences are internalization which implies the character rather than showing him/her, then by the time the reader reaches the end of that paragraph he or she will be looking hard to find the character subject from whom these internalizations are coming. Like this:
Where were the diamonds? This place wouldn't be safe for long, for sure and certain. Garmin's feet crept quietly across the floor.
Your reader might not realize it, but he/she has been looking for Garmin. But you haven't provided him for the reader; you've only provided his feet. Through the power of the grammatical subject, your reader's eyeballs have been transformed so all they can do is give the fish-eye view of what's going on. Not only will it give a strange feeling of an exceedingly close view of disembodied feet, but the reader may experience uncertainty about whether Garmin is really the character he/she is looking for. If Garmin's your protagonist, this is not a good thing.
Casting a glamour on readers
This one is a broader extension of the last power. When a fairy casts a glamour, the victim can't see what's real. When you choose not to put your protagonist as the subject of the sentence, you're deliberately making that person less visible. If you put your subject after a long "when," "before," "as," etc. clause, you're hiding your subject behind a screen. If you provide a body part, or a piece of clothing, or other evidence of the character's movements as subject, it will make the reader feel far from the character as if they're observing them externally (often from the fish-eye view!). If you choose to put your protagonist in the grammatical object position, you're making him/her into a victim and someone or something else into the position of agent/actor/do-er. If beta readers tell you your protagonist isn't ever acting or taking initiative, check to make sure he/she isn't spending too much time outside the subject position. Simply putting the protagonist in subject position isn't going to make him/her into a strong, pro-active character necessarily, but it's a step in the right direction. Furthermore, when people talk about not using passive verb forms, they may actually just mean that objects or thoughts or ideas or body parts are spending too much time in subject position in your story, rather than THE ACTOR, the protagonist, the one who should have primary place there. The flip side of this, of course, is that if you want to make someone invisible - such as when your protagonist discovers some terrible crime has been committed by an unknown (invisible!) agent - in those cases you should use the passive for whoever committed it. If your protagonist did it, but is in denial about having done it, one way of expressing this would be to have that person think about the act without placing him/herself in subject position, using passive instead. This is done deliberately in politics all the time, because by speaking in passives, politicians cast a glamour over their listeners and make invisible the actors behind critical events. It's a great tool for writers, too - but when you're dealing with your own protagonist, perhaps you can see why making the main character invisible (or distorting our vision of him/her) isn't such a good idea.
I'm sure there would be more I could say on this topic, but I think this is as much as I can fit into the extended metaphor this morning! I hope you find it interesting and helpful while you consider drafting and revisions.