SUM: You use this function when you want to aggregate the numbers based on the list formatted line item in the source module. This is considered to be Many to One relationship - meaning numbers from multiple items will be consolidated and the result will be shown in one single item.
LOOKUP: (Used in the scenario when there is a mismatch in dimensionality) You use this function when you want to get attribute(s) or numbers related to the list which is not the dimension in your target module. Similar to what excel does but in Anaplan we have concept of dimensionality. It is also used in the Top down approach (i.e., Cascading the values from Parent to Child)
SELECT: (Used in the scenario when there is a mismatch in dimensionality) You use this function when you want to get the data of any particular list member of the source module which is not the dimension in target module. This function is no longer being used on Lists anymore due to Scalability issues. It is restricted to Versions and some Time Periods Like (YTD,YTG and All Periods)