The most common reason for interception is that an aircraft is not communicating with ATC and is somewhere that they aren't supposed to be. Often times it turns out to be unintentional with the aircraft on a VFR flight (i.e. poor flight planning or deviating for weather) and straying into a restricted or prohibited area.
In these cases, the standard intercept signals (AIM 5-6-4) are designed to attract the pilot's attention and have them proceed out of the area. In addition to the "standard" signals, they can also drop flares in front of the airplane to really get their attention if needed.
In cases where the airplane still doesn't respond and the aircraft appears to be a threat to national security, the decision can be made by a few senior military and civilian officials to actually shoot down the aircraft.
For more detail, in the US section 5-6-2 of the AIM gives the following reasons that a civil aircraft may be intercepted by a military aircraft:
5-6-2. Interception Procedures
1. In conjunction with the FAA, Air Defense Sectors monitor air traffic and could order an intercept in the interest of national
security or defense. Intercepts during peacetime operations are vastly
different than those conducted under increased states of readiness.
The interceptors may be fighters or rotary wing aircraft. The reasons
for aircraft intercept include, but are not limited to:
(a) Identify an aircraft;
(b) Track an aircraft;
(c) Inspect an aircraft;
(d) Divert an aircraft;
(e) Establish communications with an aircraft.
Flightgear.org also has a good wiki on interception procedures that covers some of the common reasons that they are used (and includes all of the ones that I was going to include):
Reasons for interception in real life
While interception of civilian aircraft is a last resort, interception
is often the only means available to identify an aircraft that have
not filed a flight plan and/or have no transponder and can not be
contacted. Apart from identification interception is as well often the
only means to redirect an aircraft that is straying into limited
airspace or is believed to be involved in illegal activities.
Visual identification of aircraft that can not otherwise be identified.
An aircraft may be intercepted and through visual signals or radio communication on emergency channels be requested to
change route and possibly to land at an specific airport if an aircraft
- is straying away from a route,
- are entering a danger, restricted or prohibited area,
- are suspected to fly illegally or is smuggling goods or persons,
- enters a countries airspace without permit an fails to follow instructions to leave the airspace or land at a specific airport,
- enters a countries airspace at different positions or routes than permitted, or
- is a hazard to other aircraft