Who is the player?

The player, aka "Frisk" is the one who is controlling the character, in your case you, in my case me. The game is breaking the fourth wall in a way.

You play as Frisk, the purple and blue striped child of indeterminate race and gender. Frisk is the character who you play as and, unless we really want to get philosophical about things, the player is Frisk.

Chara, the fallen child, or whatever you named them, is not the player. You never control them- all you do is name them.

Naming a character who is not you is nothing unique. You name your rival in the Pokemon series, yet I've never seen anyone get confused and think they're Gary Oak. The only difference is that Undertale misleads you and, in its defense, it never said whose name you were entering.

Flowey is talking to you, the person playing the game, not a separate character within the game. You have the power to save and load the game, just like Flowey. You have the power to reset it. This is more or less the point of the story; it's a deconstruction of typical save/load mechanics and their implications to the inhabitants of the world, illustrated by someone (Flowey) wielding that kind of power over the world and then of someone else (you) overriding it.

It seems that he's calling you by the name you entered under the assumption that you entered your own name. That was, at one point, Toby Fox's casual recommendation. https://twitter.com/tobyfox/status/643945593308618752

Some people say "What am I supposed to name the fallen child?"

Your own name.

This was followed immediately by,

I mean, if you can't think of anything else lol

As for who you're actually naming, you are correct that it is the first fallen human. They are referred to by fans as "Chara", because the name entry screen responds to that name by calling it "true name" and because erasing the name in the save file results in it being filled in with "Chara". Aside from that, the game doesn't acknowledge or imply Chara's name. Presumably, it was discovered by editing the save file or by examining the game's data file.

The way that I interpret the pacifist ending is that after Frisk/Chara restores everyone's memories by "reaching for their souls", he/she then finds his/her own soul (which Asgore would have taken when he/she died) and recovers his/her own memories. Chara now fully realizes that he/she is the same person as the first human. After the Asrial fight, Chara sees that Asrial is coming to terms with grief. Because of what had happened before, where Asrial put everyone's lives in danger, Chara does not want to reveal who he/she is, as it would freak everyone out and make Asrial return to his obsessive self. Chara decides to use the fake name "Frisk", to protect everyone.

There is a lot of confusion between Frisk and Chara, the canonical name of the first fallen human. In most of the game, the player that you control yourself is Frisk, and always has been Frisk. Chara, on the other hand, is the first fallen human and the childhood friend of Asriel Dreemurr. There are two answers for your question that I've seen before, and I'm not sure which is correct.

1. The character you name at the beginning of the game has zero connection with you as a person, and was cleverly used by Toby Fox as deception throughout the game. When Flowey talks to you at the end, since you probably put your own name in when naming the Fallen Human, Flowey calls you what you entered.

2. Chara is the in-game representation of what you are, and what your desires were, mainly throughout the genocide route of the game. This was why Toby Fox persuaded you to put in your own name at the beginning of the game.

All in all, it's clear that you and Frisk are separate entities. While you may control Frisk, you are not one in the same with them. Flowey mentions before doing a True Reset that you should let Frisk be with their family and who they love.

Most say that the Player is Frisk But - To be entirely honest, there is evidence that the Player is neither CHARA nor Frisk, but the Player was teaching CHARA this entire time. In the beginning, you wake up on a bed of flowers. Maybe that's just Frisk regaining consciousness after the fall. Or maybe that's when CHARA finds the human and controls the soul immediately. It's whenever you die, someone tells you to stay determined, using whatever you name the Fallen Human.

"CHARA, stay determined." -Frisk

In the Genocide Run, CHARA does speak to someone. On one hand - they could be speaking to the Player. Or they could be speaking to Frisk.

You're Chara, right? -Flowey

It's me, CHARA -CHARA

"Since when were you the one in control?" -CHARA

This could be evidence that Frisk was never in control when first waking up. Sure, the window does crash. That's possibly because CHARA is rebellious and doesn't need you to teach them anymore. In the end of the True Pacifist Run, CHARA (Who is guided by the Player), decides to sacrifice the credit to Frisk to let them live their life. They were taught to be kind, so they decided it's time to be kind to Frisk as well.

In the end, it's up to the Player to teach Chara to be kind or to murder.

There's an important point I think all the other answers are missing. The only person in the game to call the player by "Chara" (or whatever name you entered) is Flowey, who we find out is actually Asriel Dreemurr.

Chara was Asriel's adopted sibling, and as Flowey, Asriel was a bit, well, insane. So it's not surprising that he confused Frisk for Chara, given that they were both human children.