You will only 'claim - BRAND' a property under your profile when it is your OWN listing. If either you made a mistake or you lost the listing, then you need to 'unclaim' the listing so the real listing agent can brand that with him/her.
You can go to my profile, click on my listings and see how it looks when a listing is branded (claim) by an agent.
There is currently one problem with one of my listings where Trulia is not recognizing the thumb-nail picture and lost the 'branding'. I emailed them to fix that.
I assume you're referring to the "Unclaim listing" text that appears on the My Listing page for a listing that you have branded using Brand Your Listings. (More here for people who are curious -- http://www.trulia.com/agents/brand_your_listings/ ).
By branding the listing to begin with, you've gotten a number of benefits:
1. Your name and photo on the listing.
2. This listing and its stats on your "My Listings" page.
3. This listing and any others you have branded show on your public profile page.
4. You get a weekly email with stats on this listing aggregated with any other listings you've branded.
If you "unclaim" the listing, these all go away. Unclaiming, however, would be the appropriate thing to do if you are no longer the listing agent for that listing, or it is no longer on the market.