Adding a 2nd Domain on top

I have a current DNS right now. I wanted to buy an account on Dreamhost and take advantage of the free DNS but can I have both domain names point to the same website?

Is there any added cost or can I simply just change the name servers?

Domain forwarding is probably the answer you are after. You can have unlimited domain forwarding on DreamHost plans without extra charge.
All the best -

Or mirroring. It depends on whether you want the second domain to redirect to the first, or have both work identically but leave the user in the URL they typed instead of redirecting to another one.

– Dan