"Archiving" is a bit like an overflow car park. When the main one gets "full", you need somewhere else to park new cars as they arrive. The difference with an archive is that you move the cars that are already parked but that haven't moved for a while; you shunt these off to the overflow car park (archive) so that the new arrivals can be more easily accessed, nearer the front gate.
Cars (data) parked in the overflow (archive) are generally ignored, except when you want to search them.
"Retention" applies to both car parks - regular and overflow - and is when you start issuing parking tickets (or deleting data). This may be simple housekeeping (Does it really matter anymore that "Fred" accessed a particular web page on your site three years ago?) or may be influenced by accountancy or even legal requirements.
You don't have to use archiving; if you have enough storage (parking space) and you still get reasonable performance from your queries even with many years of accumulated data, then you don't need archiving (and well done; you database structure must be pretty good). It is, however, a very common method used to streamline systems by moving the bulk of records, that aren't used every day, into secondary storage. For example, taking new orders from a customer today probably doesn't need to go through their entire, twenty year transaction history but a once-a-year, Management Information exercise to identify long-term customers would.