What exactly is "campus security"?
Many universities (both public and private) in the US have campus police.  These are actual police officers who can carry guns and have the power to arrest someone or write a citation.  On a large campus with tens of thousands of students, this can be a quite sizable police force.  The campus police typically have jurisdiction limited to the university campus, but in some cases, they share jurisdiction with local police in the area near campus.  

Other universities have campus security.  These are security guards who operate much like the security guards you might find at a sporting event or concert.  They don't have full police powers, but they can certainly remove someone from campus who is misbehaving.  When an incident becomes serious, they'll call in the local police force.    

"Security" is often used as a generic name for these forces, whether or not they have actual police powers.  On many campuses, there is a mixture of police (more highly trained and paid more) and security guards (less trained and less expensive) that are assigned to deal with different kinds of situations.  

So, 


Not exactly.  Campus police have full police powers (to arrest, use 
deadly force, etc.)  Campus security typically has less authority.  
Campus police or security are typically the first responders for criminal activity (robbery, assault, etc.)  They enforce campus rules that may not carry criminal penalties.  Campus security can call in the local police for more serious incidents.   Anyone on campus (students, faculty, or staff) can call in the campus police or security, although making a frivolous complaint could get you in trouble.    
Unless there's some jurisdiction agreement between campus police and the regular local police force, it wouldn't be appropriate to call the campus police to respond to an incident off campus- the campus police would transfer your call to the appropriate police department.  Note that "campus" might include all university-owned buildings, even if they're physically distant from the main campus.  In particular, university-owned housing is typically "on campus."  


At another level, most campuses in the US have their own set of rules for student conduct and an administrative process for judging students who are charged with violating the rules.  This is separate from the legal system.  It gives some leeway to university administrators to deal with infractions that might be crimes if they were handled by the regular legal system.  This can be controversial, particularly when it extends to more serious offenses such as sexual assault. 

P.S.  

Some large campuses in the US also have their own staff to provide firefighting and emergency medical assistance.  When there are science and engineering labs, it's particularly important that firefighters be trained in how to deal with emergencies involving hazardous materials.     

Many campuses in urban areas have free escort services- a student (or even a staff or faculty member) calls up security to request an escort to walk with them (or give a ride in a car) from one part of campus to another. You can't expect instant service, but a typical wait time might be 20 minutes.   The escort might be provided by a campus police officer who is already out on patrol, but it's more common for there to be a group of non-police employees and vehicles dedicated to this. These services are mostly used late at night- if you've been working in your lab until 1 am, you might want an escort back to your dormitory or the parking garage.       

In the US, the Clery Act requires colleges and universities to prepare an annual report with statistics on crimes and other incidents (such as fires) that occur on and near campus.  Potential students often read these reports to determine whether a campus is "safe."   This has made campus safety an important concern to administrators.