In the state of Nevada:
Under NRS 207.200, a person can be found guilty of a criminal trespass upon land under three distinct legal theories. This means that there are three ways for the prosecution to prove a trespass.
First, pursuant to NRS 207.200(1)(a), a person commits a trespass when he goes onto another person's property "with intent to vex or annoy the owner or occupant" That is, if the prosecutor can prove that a person went onto another person's property with the specific goal of vexing or annoying the property owner, then the person can be convicted of a criminal trespass. Similar to any charge where intent is at issue, the prosecutor would have to show the defendant's "intent to vex or annoy" through presenting circumstantial evidence. This would typically be proven by explaining that the defendant had an antagonistic relationship with the property owner, or by showing that the defendant's actions when on the property were, in fact, vexing or annoying.
Second, also under NRS 207.200(1)(a), a person commits a criminal trespass when he goes onto another person's property to "commit any unlawful act." Typically, this theory is used if a person goes onto someone else's property to solicit prostitution, use illegal drugs, or commit some other minor offense. However, keep in mind, if the "unlawful act" is a felony or a theft, then the person would be guilty not of a mere trespass, but a burglary, which is a more serious offense.
Third, under NRS 207.200(1)(b), a person commits criminal trespass if he goes onto someone else's property "after having been warned by the owner or occupant… not to trespass." In other words, if a property owner clearly tells you not to enter onto his property, and you willfully go onto that person's property anyway, then you are guilty of criminal trespass. This is where the idea of being "trespassed" becomes relevant. When a property owner informs you that he wants you off his property, you have been trespassed and can't return to the property without committing a criminal trespass. In Las Vegas casinos, this happens frequently. If a person has too much to drink, is suspected of committing a crime, or otherwise causes problems, then casino security will record that person's identity, and actually give the person a letter indicating that he is banned from casino property and all other properties around town owned by the same company. If the person returns to the casino, he can be arrested for trespassing. Under NRS 207.200(2), a "warning" against trespassing can be given by posting a "no trespassing" sign, fencing in a certain area of the land, or giving an oral or written demand to leave the property.
Keep in mind no gun signs carry no weight of law here.