What does it actually mean by the error “Arithmetic expressions must use numeric arguments”in Salesforce?

You nearly made me forget the syntax there !! I've corrected the trigger above, try that to see if it works.

You need to have a for loop to iterate over all the records that have invoked the trigger.

Also a before trigger makes more sense, since you're manipulating values on the triggering records.

trigger Peachtree_items_process on Peachtree_items__c (before update, before insert) {

for( PeachTree_Items__c peachTreeItem : trigger.new) //iterate over all items
if (peachTreeItem.Stock_Minimum__c != null  && peachTreeItem.Qty_On_Hand__c != null &&
peachTreeItem.Qty_On_PO__c != null) {
peachTreeItem.Stock_Reorder_Qty__c =
peachTreeItem.Stock_Minimum__c -
(peachTreeItem.Qty_On_Hand__c + peachTreeItem.Qty_On_PO__c);
}
}