A Lot Number is used to define a batch Output from Production, or a group of an Item upon Receipt. This is largely used for Item Tracing and costing. Sometimes you will only have a single Lot Number in inventory for an Item, at other times multiple Lot Numbers in inventory for the same Item.
A Serial Number is similar to a Lot No, except it represents a quantity of 1. Where a Lot Number represents a group of an Item, the Serial Number is specific to a single Unit of that Item. Again, largely used for Item Tracing and Costing.
A Package Number is used to define the contents of a box. It could be a single Item or multiple Items in a box.
In the warehouse you may have a single Item with multiple Lot Numbers assigned. The Package No. allows me to put all of that product into a single Package (aka box), which then knows the various Lot Number information of the Items inside. This is the same concept as a License Plate, available through ISV Extensions.
It is possible that your Package Number contains a single Item, and that Item has a single Lot Number in the box. In that case, they are similar, but obviously that is just a single example.
Hopefully this helps clarify the differences.
Best Regards,
Ben Baxter
Accent Software Inc