How do I get the name of a player?

Every object in roblox has a .Name property, which will give you a string of the object's name. Text labels have a .Text property that can take a string and show it on the label. If you had a LocalScript inside of the text label, you could get the player who "owns" the text label through LocalPlayer and set the text to their name:

script.Parent.Text = game.Players.LocalPlayer.Name

Since it is fairly simple, I will post a quick example of it

-- [Declaration Section]
local Player = game:GetService("Players").LocalPlayer;
local PlayerName = Player.Name;
local PlayerGui = Player:WaitForChild("PlayerGui");

-- //TextLabel Location 
local ScreenGui = PlayerGui:WaitForChild("ScreenGui");
local TextLabel = ScreenGui:WaitForChild("TextLabel");

-- [Output Section]
TextLabel.Text = PlayerName;

Explanation

The following script is a LocalScript. First we are declaring the Local Player. Then we are getting its name through Player.Name. Name is a roblox identifier which essentially gives us the Name of the object we are referencing it to.

Since in our case we want the Name of the Local Player, we are going to do Player.Name. Then we implement the Name in the TextLabel. We do the following by adding TextLabel.Text = PlayerName.

Should you have any questions, feel free to comment below.

Hello, to get a name of a player is actually quite simple.

script.Parent.Mouse1Down:connect(function(player) --Not sure what you wanted so I'm just setting the text to whoever presses a text button 
    local gui = player.PlayerGui.ScreenGui --gets the gui
    gui.TextLabel.Text = player.Name --Sets the text to the players name
end)

The simplest way would be this:

local Player = game.Players.LocalPlayer
local TextLabel = script.Parent -- Put the location of the label here

TextLabel.Text = Player.Name

Make sure your using a LocalScript for this since each player has a different name and your dealing with GUI elements.