Finding out who someone really is on a forum depends on several factors. The moderator of the forum may be able to find out more information than just another user of the forum. It also depends on why the person on the forum's identity is being sought. If the person is engaging in criminal behaviour, harassment, threats or admitting to crimes, then the person can be reported to the authorities who will then use his IP address to find out who he is. Most ISP (Internet Service Providers) will not divulge the names of clients without being asked to by authorities.
Ask the person in a private "PM" message who he is. If the context of the identity search is positive and friendly, then it should be acceptable to contact the user privately and ask for more personal information, such as a name and e-mail. For example, a user on an RC aeroplane forum may see another person posting who lives in his area. That user may want to contact him to meet up in person and fly planes together.
Look at the registration details of the user if you are a forum moderator. When the user signed up for an account on the forum she filled out a registration form. Look at the form; this will likely contain an e-mail address, IP address (which is logged by the site, not the user) and may contain other information such as a name. From this information the identity of a person on the forum may be determined. For example, if the user "ABC123" registered with an e-mail "firstname.lastname@example.org" then the user's identity could be assumed to be John Smith.
Report the user to the moderator of the forum, if you are not the forum moderator. Likely the user is engaging in some sort of negative behaviour if that user simply can not be asked for his identity. Privately message the moderator and give the username of the person in question and any supporting documentation as to why his identity should be revealed. For example, this could include threats in posts or other negative behaviour.
Report the user to the police. This can be done by both the moderator and a regular user. If the user in question is breaking the law in any way; libel, harassment, threats, or admitting to or soliciting illegal acts, then the authorities can be called to investigate. The authorities will contact the forum moderator or webmaster and obtain the IP address of the user (if necessary by court order.) They will then track down the service provider of that user and connect the IP with a customer. The police will contact the account holder and question him about the behaviour on the forum. The police may ask the moderator or other forum users to testify about the behaviour of that individual.