Disclaimer--Not legal advice.
Now that that is out of the way... If you're properly served and don't make an appearance in the case (i.e., don't file an answer to the complaint), then you're not in contempt of court, but the plaintiff CAN take a default judgment against you. That means the plaintiff gets a judgment for basically whatever damages he or she can prove up to the court. There are limited grounds upon which a default judgment can be overturned. Some of these grounds would be lack of subject matter or personal jurisdiction, improper service, etc.
If the case was filed in Federal court, then typically the plaintiff will start out by sending a request for waiver of citation pursuant to Fed. Rule of Civ. Pro. 4. If you don't waive service (you'd generally have 60 days to do so), then the P can have you served in a traditional fashion. The Court can also approve alternative methods of service. (In some internet-related actions, the Court has actually approved service of process via email!)
The procedural rules of the various states will vary somewhat from the Federal rules, but I'm aware of no states where failure to appear upon being served with process will result in a finding of contempt. Service of process is simply a procedural method to ensure you have adequate notice of the case, and is not an order to appear.
On the other hand, if you're served with a subpoena to testify, either for a deposition or in court, and fail to appear, then you can be found in contempt and you can be fined, thrown in jail, or have your pleadings stricken, etc., within the court's discretion. Unlike service of process, a subpoena IS an order of the court to appear, and you can be in contempt for not complying with it.
Hope that helps.
P.S. This is not legal advice and there is no attorney-client relationship by reason of this post, AND this post is made on the express understanding that no one reading this thread will be relying on anything contained herein!
[ August 01, 2001: Message edited by: jdeets ]