In general, my suggestions work well if you think of the whole thing as a big semi-secret organization. Use words that refer to people's positions within an organization - President, VP, etc. It may sound cheezy, but it works!

Vampire

Recommended replacements: Kindred, a member of our Organization
Usage: "Are you Kindred (a member of our Organization)?" rather than "Are you a Vampire?"

Ghoul

Recommended replacements: Assistant
Usage: "I'll have my Assistant take care of that." rather than "My ghoul will do that."

Camarilla (or Sabbat, or Anarchs)

Recommended replacements: our Organization (or just don't refer to it)

Traditions (including the specific names of them)

Recommended replacements: our Code, our Bylaws
Usage: "That is a clear violation of our 4th Bylaw (section 4 of our code)!" rather than "You just broke the 4th Tradition!!!"

The Embrace, Embracing someone

Recommended replacements: Recruitment/Recruiting, Offer Membership
Usage: "Now there is someone we could offer membership!" rather than "I'd like to Embrace him/her!"

Elysium

Recommended replacements: Meeting site, or simply refer to it as what it is
Usage: "This _IS_ the Student Center, you know!" rather than "We're in Elysium!"

Clan

Recommended replacements: Family, Line, Lineage
Usage: "Which family (or line) do you claim?" or "Which family (or line) claims you?" rather than "What clan are you in?"

specific clan names

Recommended replacements: each clan has a nickname - Brujah=Scholars, Gangrel=Beasts or Wolves, Malkavian=of the Moon, Nosferatu=Hidden, Toreador=Rose, Tremere=???, Ventrue=Kings. In some of these, you can use them as surnames.
Usage: "I am a scholar." rather than "I'm Brujah." "I'm descended from the Rose family." rather than "I'm a Toreador." Some of these examples work better than others. Be creative!

names of Disciplines

Recommended replacements: the ability to (fill in the blank)
Usage: "He used his ability to persuade people on that poor mortal." rather than "He Dominated that poor mortal."
On a related note, as Alex pointed out to me, few Kindred would refer to their disciplines by the game names. They would refer to them both in and out of Elysium by what they give you the ability to do.

Prince

Recommended replacements: many, dependent upon your personality - our Leader, our President, the Boss, the Man, etc.
Usage: "Let me introduce you to the esteemed President of our Organization." rather than "Let me introduce you to the Prince."

Primogen

Recommended replacements: representative of our Family, head of our Committee, advisor to the (pick your word for Prince)
Usage: "I am Ophelia Selby, representative of the descendants of the Rose family." rather than "Ophelia Selby, Toreador Primogen."

Seneschal

Recommended replacements: Vice-President, (pick your word for Prince)'s assistant, etc.

Keeper of Elysium

Recommended replacements: this one is hard... I dunno!

Sheriff

Recommended replacements: Sergeant at Arms (if you're going with the organizational structure, this works)

Chief Harpy, Harpies

Recommended replacements: Gossips, the In-Crowd, Social Butterfly
Usage: "Oh, that's Sigrid, the head of the In-Crowd." rather than "Oh, that's Sigrid, the Chief Harpy."

Neonates, Ancilla, Elders

Recommended replacements: none, really. These are more along the lines of game terms that we use, and they're all subjective anyway! On a related note, it's not a very good idea to say "Oh, I've been like this for several hundred years." That makes it kinda obvious you're not normal.

References to products created by White Wolf or other companies are not challenges to their copyrights.