Attempt to Index field 'Character' a nil value?

The problem here is that the character hasn't loaded yet. When you get the character, check if the character exists, and if it doesn't you need to wait for CharacterAdded.

local player = game.Players.LocalPlayer
local char = player.Char or player.CharacterAdded:Wait()
local runanim = char:WaitForChild("Humanoid"):LoadAnimation(script.Running)

uis.InputBegan:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift then
        char.Humanoid.WalkSpeed = 20
        runanim.Looped = true
        runanim:Play()
    end
end)

uis.InputEnded:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift then
        char.Humanoid.WalkSpeed = 16
        runanim.Looped = false
        runanim:Stop()
    end
end)

Not related to your question, but you should also make sure that when the shift key is held to make a capital letter in the chat, it doesn't trigger the running animation.

local player = game.Players.LocalPlayer
local char = player.Char or player.CharacterAdded:Wait()
local runanim = char:WaitForChild("Humanoid"):LoadAnimation(script.Running)

uis.InputBegan:Connect(function(key, gp)
    if key.KeyCode == Enum.KeyCode.LeftShift and not gp then
        char.Humanoid.WalkSpeed = 20
        runanim.Looped = true
        runanim:Play()
    end
end)

uis.InputEnded:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift then
        char.Humanoid.WalkSpeed = 16
        runanim.Looped = false
        runanim:Stop()
    end
end)

From what i'm seeing here you don't have a player variable to call "plr", and that's why it returns the index as nil. If you put the script inside of the StarterCharacterScripts folder, ( which is inside of StarterPlayer ), you can get the Character only by using script.Parent, since the script will be parented to it as soon as the character is loaded, which is what i did here on this script. I renamed and separated some variables just to make it easier to understand. Hope this helps!

UIS = game:GetService("UserInputService") 
local Character = script.Parent
local Humanoid = Character.Humanoid
local Anim = script:WaitForChild("Running") 
local Track  = Humanoid:LoadAnimation(Anim)

UIS.InputBegan:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift then
print("pressed!")
        Humanoid.WalkSpeed = 20
        Track.Looped = true
        Track:Play()
    end
end)

UIS.InputEnded:Connect(function(key)
    if key.KeyCode == Enum.KeyCode.LeftShift then
print("unpressed!")
        Humanoid.WalkSpeed = 16
        Track.Looped = false
        Track:Stop()
    end
end)

so basically you dont have a plr variable which you can get character from. or thats not your entire script.

im guessing your using a local script so this is what you would have to do to get the plr:

local runanim = game.Players.LocalPlayer.Character:WaitForChild("Humanoid"):LoadAnimation(script.Running)
local plr = game.Players.LocalPlayer

    uis.InputBegan:Connect(function(key)
        if key.KeyCode == Enum.KeyCode.LeftShift then
            plr.Character.Humanoid.WalkSpeed = 20
            runanim.Looped = true
            runanim:Play()
        end
    end)

    uis.InputEnded:Connect(function(key)
        if key.KeyCode == Enum.KeyCode.LeftShift then
            plr.Character.Humanoid.WalkSpeed = 16
            runanim.Looped = false
            runanim:Stop()
        end
    end)

and that would get the plr for you and the Character.