0) Somebody allegedly uses property for an illegal purpose. By law, they are deemed to have transferred title to the United States Government by dint of the illegal activity (if in fact the illegal activity can be proved).
1) The government files for a seizure warrant in US District Court. The owner of the property (here, the domain) does not get a say, nor any notice that this is happening.
2) The government seizes the property and provides notice to the owner, if known, and any person who might have a claim on it. For example, if the property is a car with a bank lien, they must notify the owner and the bank.
3a) The government files a complaint for forfeiture in US District Court (or in state court). This is called an "in rem" action--meaning that it's not a lawsuit against an individual, but a suit to determine title to property. The United States claims that it owns the property because of the transfer-by-law that occurred at zero, supra. Anybody who disagrees can stake their claim. The judge determines who gets the stuff.
3b) The government doesn't file anything, and the owner sues the government for a civil rights violation by unlawful taking of property without due process. The suit proceeds as above.
The cases determining whether due process has to occur pre-seizure or post-seizure are complicated, and beyond the scope of this author's knowledge or this post.
For reference, I am a lawyer and have posted this explanation based on my legal study, but it should be considered scholarship (information for general knowledge) and not legal advice (information specific to an individual's problems). If you are in need of legal advice, you should consult a qualified lawyer in your jurisdiction.