OK - but it's complicated! ;) Package can be 'the thing' you send. It can also be the verb (in several forms) meaning 'to make a package'. Packing can be the present participle of 'to pack' meaning to put things into a bag or package. It can also be a noun meaning materials used to pack things, similar to 'padding'. Packaging can be the present participle of 'to package'. It can also be a noun, meaning the materials you use to make a package with. Please send this package to my brother. I need to package up these sheets and send them to my sister. I am packing my bags for my trip to Vietnam. Have you got any packing I can put in this box with these bottles? He is packaging up your fish right now. We will sent it as soon as possible. Have you got any packaging that I can send these bottles in? Sometimes people use the nouns packing and packaging to both mean the materials you put in as padding.

First, we have the verb pack which means putting a wrap in something, fill something (like a suitcase) with objects or put something in a box. So, packing is a conjugation of this verb. A package is the actual object which is wrapped or packed. Finally, packaging is the material we use as a wrap or to put a package on. For example, let's say you order shoes online. The company who is going to send it to you must put them in a box so there will be a person [packing] your shoes when they are about to send it. Then, when they arrive at your house you will receive the [package] from the mailman. When you want to use your shoes you will take them out from the [packaging] they arrived on.