You have three if statements, but you only end one of them.
What you probably want here is else if:
display dialog "Open Which Application" buttons {"Chrome", "AppleScript", "Textedit"}
set the button_pressed to the button returned of the result
if the button_pressed is "Chrome" then
open application "Google Chrome" -- action for 1st button goes here
else if the button_pressed is "Applescript" then
open application "AppleScript Editor" -- action for 2nd button goes here
else if the button_pressed is "Textedit" then
open application "TextEdit" -- action for 3rd button goes here
end if
(Also, you're really supposed to use end if, not just end, but AppleScript Editor will fix that for you.)
Alternatively, you could end each one:
display dialog "Open Which Application" buttons {"Chrome", "AppleScript", "Textedit"}
set the button_pressed to the button returned of the result
if the button_pressed is "Chrome" then
open application "Google Chrome" -- action for 1st button goes here
end if
if the button_pressed is "Applescript" then
open application "AppleScript Editor" -- action for 2nd button goes here
end if
if the button_pressed is "Textedit" then
open application "TextEdit" -- action for 3rd button goes here
end if
However, the cases are obviously mutually exclusive, so there's not reason not to use else if instead.
If it helps, type the lines in one by one and see how AppleScript Editor is indenting them:
display dialog "Open Which Application" buttons {"Chrome", "AppleScript", "Textedit"}
set the button_pressed to the button returned of the result
if the button_pressed is "Chrome" then
Open application "Google Chrome"-- action for 1st button goes here
if the button_pressed is "Applescript" then
Open application "Applescript Editor"-- action for 2nd button goes here
if the button_pressed is "Textedit" then
Open application "Textedit"-- action for 3rd button goes here
end
That should make it obvious what's wrong.