In many states, the sheriff of the county will serve papers. But there is a cost associated with this. The sheriff completes the certificate of service, and then, depending on the jurisdiction, either you or the sheriff can file the certificate with the court. In some states, it is possible to find private process-servers, but of course, they must be paid as well. Another alternative is by legal notice in legal journal or a paper of mass circulation. Of course, as the first response stated, each jurisdiction has its own rules of service, and the best course of action is to consult an attorney who practices in your county. In some counties, there are legal services available at little or no cost for people who are below a certain income level, so that is another avenue that might be worth exploring. Additionally, some local bar associations will provide a list of attorneys who are willing to provide a brief consultation for a reduced fee, another option that might be available to you. I have provided a link for you to help you find out if there are any free legal services available to you in your state. Good luck!
As is the case with most legal stuff, the answer to this is going to depend a lot on where you live, and possibly even on what kind of law suit you are involved with.
Just to give you some idea of what might be possible, in California, you can have the server do what's called "substituted service." They can do this if they've made a good faith effort to actually serve the person but haven't managed it.
This process involves A) giving the papers to an adult at the defendant's legal residence, B) sending the papers to the defendant and C) filling out statements as to what the server tried to do to actually serve the defendant in person.
Most states have rules for what you can do if you can't find the defendant at all.
But for information about your own state and what is applicable to your case, you really ought to consult a lawyer.