Anonymity allows Internet forum users more freedom to express their views, but sometimes people abuse that freedom. If somebody on a forum refuses to reveal his identity, moderators or administrators of the forum will usually be able to access data that at least gives a clue as to the identity. In some cases, though, you won't be able to suss out an identity without the help of law enforcement authorities.
Most forums ask for a username, email address and password when somebody signs up. While passwords should always be secret, moderators should be able to access email addresses, even if the user has chosen to hide this in his public profile. In some cases the email address may contain the user's real name. In others you may be able to search online for the address to see if it is mentioned on any other site that might reveal the owner's identity.
Moderators should be able to access the IP address of the computer a forum member used, either when registering, on her latest post, or both. You can check the geographic location that matches this address at sites such as IP Location (see Resources), though bear in mind that in some cases this will be the location of the person's Internet service provider. If you can get a geographic location, it may help narrow the search. Alternatively, if you suspect the person has previously posted on the site under a different username (and you know that user's real identity), you can compare the IP addresses on the accounts.
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ISPs & Police
Internet service providers will have records that allow them to match an IP address with an individual customer, or at least the customer's household. If an IP check reveals a user's ISP, you can ask the ISP to provide his identity, though it's unlikely the ISP will comply. In most cases an ISP will only do so upon a request from law enforcement authorities and/or a court order. If your reason for wanting to know a user's identity relates to potential criminal activity such as online abuse or harassment, consider contacting the police and letting them work on identifying the individual.
If you cannot get access to forum data, or it doesn't bring anything up, you can try a few Hail Mary plays. These all might throw up a possible match, but certainly won't be conclusive. If the person has a distinctive avatar, try using Google Image Search to see if she has used it elsewhere. Similarly, carry out a text search if she has a distinctive quotation or other text in their signature, which they may have used on another forum or social site where her identity is clearer. Finally, try right-clicking on any images the user has posted; she may have slipped up and linked to an image hosted on her own site or an identifiable account on another site.