For your specific itinerary it's a little unique, and will vary in each direction.
All passengers travelling to the US will need to pass through US immigration and customs, even if they are only in transit. See this answer for some details on why that is.
On your outbound trip (YVR > EWR > PMI), you will pass through US immigration and customs whilst still physically in Canada, at Vancouver airport, under a program called Pre-Clearance. You do not need to physically carry your bags through customs, but they will be held by US immigration there until they allow you to pass through to the boarding area (at which point are have officially "entered" the USA).
When you arrive in Newark from Vancouver, you arrive basically as a US domestic passenger would. As a result, your bags will be automatically transferred to your international flight without you having to collect them. You will not need to pass through immigration or security, you can simply go directly to your gate. (Depending on the airport you may need to go through security if it is not possible to get from your arrival gate to your departure gate without doing so, but I don't believe that's required for AC/UA at EWR).
On the way back (PMI > EWR > YVR) it's a different story. You bags will be tagged all the way to your final destination, however you will need to pass through immigration and customs on arrival in the US, which means you will need to collect your bags at Newark, walk them through customs, and then re-check them at the baggage drop area which is just outside of customs. From there you can head through security to your gate for departure.
Note that as you will be entering the US (in both directions) you will need to have the legal right to do so. If you're a Canadian citizen then you're generally going to be fine there, but if you're not then you will require either an ESTA or a US Visa before you will be allowed check-in for your initial flight.
