Instead of suing you they will likely file a case with the World Intellectual Property Organization (WIPO) who has authority in these types of trademark issues across international borders. It will cost the company $1200 or so plus a lawyer to file the documents with WIPO. WIPO will notify your registrar (like GoDaddy) who will in turn notify you. You will also get a letter from WIPO at the snail mail address you have registered for that domain. You can visit the WIPO website to see the official complaint and you will have a certain period of time to respond online. You don't have to pay any fees. You don't need a lawyer.
If you feel you have a case to make then you can respond with your reasons. It is good to look through the past cases on the WIPO site (they post them all) to see what kind of arguments work. Pretty interesting reading.
A WIPO judge is assigned to the case and he will look at the explanations and also do some research on his own to validate. Even if you don't file a response WIPO could decide in your favor. However, if the other party has a registered trademark and you have a domain that directly matches or you are using it in such a way that a normal person could be confused, then they will almost always decide in the trademark holder's favor. The only exceptions might be common words that some company has trademarked that could mean something else too and you are not using it for the same meaning that was trademarked.
If they decide against you, then they will email you and also your registrar. Your registrar will notify you that they will transfer the domain in so many days (15?) and when the time comes, it will be moved. However, until the new owner changes the DNS it will continue to point to your site.
Can you tell I have been through this myself?

I had one site that I lost in a WIPO dispute that the big dumb company didn't change the DNS for over 6 months and I made quite a lot of revenue on it during that time.
Anyhow, if that happens that they take your domain, there is no cost, except the lost domain registration fee. Of course they could try to sue you for damages to their reputation and then you end up paying court costs or whatever.
One thing to NOT do is to offer to sell the domain to the trademark owner as that is taken by WIPO as a sure sign of domain-squatting, which will get your domain yanked, no further evidence needed.