There is a saying "tall fences make good neighbors." For many people this could be true. However, there are some neighbors who do not respect tall fences or any other border. There are a few sayings for that type of neighbor as well. "Let's get a land survey," and "I'd hate to go to court over this." Which of those you will end up uttering depends on just how your neighbor chooses to cross your borders.
This land is your land; this land is my land...
If it is there property, a fence or other structure, or even an addition to their home that is one or crossing you property line; then you have a boundary dispute. The first step to dealing with this is to talk to your neighbor and then compare deed copies. The exact wording of the deeds should tell you where your property lines are. If they do not then you will have to agree to pay for a survey. Whether you pay for this yourself or work out a deal with the neighbor will depend on the state of your relationship with that person.
After you know exactly where your boundaries are, they should move their belongings. If not, try going to mediation. Many homeowners associations offer mediation services. If yours does not, you can hire a private mediator to help you work through the encroachment issues. Otherwise, you will have to go to court and have a judge order the guy next door - to keep his things next door.
Some neighbor disputes have a naturally occurring root - trees. The limbs hang over the fence you built or break and fall into your yard. Maybe the roots are pushing through onto your property. If the tree is on a boundary, most states will not allow either party to destroy it.
Leaves, pods, acorns and the like falling onto your property are considered a natural occurrence and are the property owner's responsibility to clear away. However, if branches fall and cause damage on your property for any reason other than a storm or act of God, your neighbor is responsible for the cleanup and damage. If the roots are pushing onto your property, they are considered an encroachment in the same manner as fences and other physical belongings. If the roots cross your property line, the tree owner has to remove it.
We don't want Rover coming over...
Perhaps it is not your neighbor's property that is crossing the line, it is their animals. A wandering tomcat or unleashed dog can leave unwanted 'gifts' or otherwise damage property. As with most cases, how you deal with this depends on where you live. Many cities have ordinances covering pets and how they should be kept and whether they need to be leashed. If the animal in question is dangerous, or has hurt someone, the owner can be held liable for any injury or damage. In some cases, neighbors can get a court order to have the animal confined.
We don't want the neighbor over either...
Last, but not least, is the most egregious and outrageous type of boundary crossing; your neighbor is what is crossing your borders. The person has cut a path through your back yard, or your front yard. If you live in a rural area, you find them hunting, fishing, picnicking, etc., on your land without your permission. There is a law that covers these situations as well. It is called trespass. Many times, it carries a criminal penalty. If the neighbor is on your property doing something particularly offensive or dangerous, calling the police is the best and most immediate way of dealing with them.
Otherwise, document their trespasses and file a police report. Get dates and times. If you can get photos or statements from people who witnesses the event, that will be of tremendous help. They can be fined and ordered to stay off your property.