what are the diff. types of functions?

By the sounds of it you're talking about events, not functions. Events are specific for each instance, you can refer to the wiki for a full list of events for any given instance.

Example:

local Brick = script.Parent

function omgPart_wuzTouched(part)
    print(part:GetFullName() .. " touched " .. Brick:GetFullName())
end

Brick.Touched:connect(omgPart_wuzTouched)

omgPart_wuzTouched <-- That's a function

print <-- That's a function

Touched <-- That's an event

connect <-- That's a method that connects a function to event so that when the event is fired the function 'omgPart_wuzTouched' is called

You can find all of the build-in (predefined) functions here: http://wiki.roblox.com/index.php?title=Function_Dump. Hope I've helped! :)

If you're looking for the functions in Lua, you're probably looking for what MidnightDeveloper said: Click here.

Else, you are probably asking what kinds of functions you can make. Functions don't have a type, you may classify a function as this and that, but from Lua's understanding, there is no types of functions.