Comets have highly elliptical orbits.  When at perihelion or closest
approach to the Sun, they are typically about the same distance from
the Sun as the Earth is.  When at aphelion or farthest distance from
the Sun, they can be well outside the orbit of Pluto.  If a comet is
observed for a sufficient period of time, its motion on the sky allows
us to estimate when it is at perihelion and how far away aphelion is
(more precisely, we can estimate the major axis of its orbit).

In 1950 Jan Oort was analyzing the comets whose orbits had been
determined.  He discovered that many comets had their aphelia at
roughly the same distance from the Sun, about 50,000 AU.  (For
reference, the Earth is at a distance of 1 AU from the Sun, Neptune is
at a distance of 40 AU, and the nearest star is at a distance of
270,000 AU.)  So Oort proposed that the Sun was surrounded by a vast
swarm of comets, stretching nearly 1/5 of the distance to the nearest
star.  

At these large distances from the Sun, these comets are only loosely
gravitationally bound to the Sun.  A slight gravitational nudge, from
a star passing within a couple of light years or so perhaps, is enough
to change their orbits dramatically.  The gravitational tug can result
in a comet either (1) becoming gravitationally unbound from the Sun
and drifting into interstellar space never to return or (2) falling
into the inner solar system.  This is the currently accepted
explanation for the origin of so-called "long-period" comets.  These
comets orbit the Sun at great distances, until a slight gravitational
nudge changes their orbit and causes them to fall into the inner solar
system, where we see them.  Because their aphelia remain at large
distances, it can take hundreds, thousands, or maybe even 1 million
years before they return to the inner solar system.  Comet Hale-Bopp
is an example of such a comet.

Theorizing that comets originate from the Oort cloud doesn't explain
the properties of all comets, however.  "Short-period" comets, those
with periods less than 200 years, have orbits in or near the
ecliptic---the plane in which the Earth and other planet orbit.
Long-period comets appear to come from all over the sky.  Short-period
comets can be explained if there is a disk of material, probably left
over from the formation of the solar system, extending from the orbit
of Neptune out to 50 AU or more.  Collisions between objects in such a
disk and gravitational tugs from the gas giants in our solar system
would be enough to cause some of the objects to fall into the inner
solar system occasionally where we would see them.  Comet Halley is
probably an example of such a comet.

Direct detection of Kuiper Belt objects occurred in the early 1990s
with the detection of 1992/QB1, see
<URL:http://www.ifa.hawaii.edu/faculty/jewitt/qb1.html>.  Additional
indirect evidence for a disk of material around the Sun comes from
images of nearby stars which have disks around them.  These disks
around other stars are several times larger than the Kuiper Belt has
thus far been observed to extend, but they might be qualitatively
similar to the Kuiper Belt.  See
<URL:http://galileo.ifa.hawaii.edu/users/jewitt/Origins-bpic.html>.

Interestingly, current theories for the origin of the Oort Cloud and
Kuiper Belt indicate that the Kuiper Belt probably formed first.  The
Kuiper Belt is the detritus from the formation of the solar system.
Objects from it that make it into the inner solar system can interact
gravitationally with the giant planets, particularly Jupiter.  Some
objects would have had their orbits changed so that they impacted with
one of the planets (like Comet Shoemaker-Levy 9 did in 1994); some
objects would be ejected from the solar system entirely; and some
objects would be kicked into very large orbits and into the Oort
cloud.