dns for re-directed subdomain

[drew354]

If someone could point me to a DNS primer that may help - I'm trying to
see if this scenario is possible -

Can I have a subdomain of site A redirect to a folder of site B (no
problem, right?)

AND

somehow rename the viewed address to reflect the domain name of site A?

So that even though I am in secondwebsite.com/subfolder

my browser shows that I am in subdomain.firstwebsite.com

If this possible I would greatly appreciate it if someone could point
me to a resource that provided detailed instructions about how to do
this.

Thanks!

Drewster