Who is the player?
This question cannot be asked without giving spoilers, so please don't step forward without truly playing it.
After completing a True Pacifist run and opening the game again
Flowey appears asking for you to leave it as is, to not reset the game, because you, "Player Name", are the only one with the power to do that.
But that confused me a lot, because we
don't name the playable character, but instead the first human to go underground (Chara), who manifests themselves and rules over everything erasing the world in a Genocide Run. But we aren't Chara, because we sell our soul to them to be able to ever play again
So... who is the player really?
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.
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.