I am trying to make an Applescript that launches a different set of apps depending on user input. However, it won't compile. What is wrong with it?
You are missing a bunch of end ifs and 2 then. 

set question to display dialog "What do you want to launch?" buttons {"Spotify", "MiniPlayer", "Remote"} default button 1
set answer to button returned of question

if answer is equal to "Spotify" then
    set myAppName to "Spotify"
    set isRunning to false
    tell application "System Events"
        if exists process myAppName then
            set isRunning to true
        end if
    end tell
    if isRunning is true then
        tell application myAppName to activate
    else
        tell application "Spotify" to activate
        tell application "Spotifree" to activate
        tell application "Spotify Notifications" to activate
    end if
end if

if answer is equal to "MiniPlayer" then
    tell application "MiniPlayer" to activate
end if

if answer is equal to "Remote" then
    set question to display dialog "What remote do you want to launch?" buttons {"IR Remote", "Android", "Both"} default button 2
    set answer to button returned of question
    if answer is equal to "IR Remote" then
        tell application SpotifyIrRemote to activate
    end if
    if answer is equal to "Android" then
        tell application RemotelessHelper to activate
    end if
    if answer is equal to "Both" then
        tell application SpotifyIrRemote to activate
        tell application RemotelessHelper to activate
    end if
end if