Why do you need to know anything about orbital elements? The short answer is that knowledge of orbit dynamics and the parameters of orbits (orbit elements) will help in understanding the behavior of satellites and knowing why they show up when and were they do. This knowledge will help you observe them.
Satellite orbits are described using a set of orbit elements stored in the form of TLE's (Two Line Elements). After some practice one can visualize a satellite orbit by merely looking at the TLE of the satellite. In this section we will introduce the orbit elements found in a TLE, and then provide some links so that further study can be made.
Here is a TLE:
MOLNIYA 1-86
1 22671U 93035A 10304.20566066 .00000807 00000-0 31628-3 0 3708
2 22671 62.2806 61.3884 7489727 276.5130 11.4192 2.03412498 06

Notice that the two lines of the TLE are numbered. The common name of the satellite, MOLNIYA 1-86, is really the zero line of the "two" line element. The common name is not really necessary as the satellite can be identified from either the NORAD Catalog number, 22671, found in both lines, or the International Designation, 93035A, found in line 1.
Six numbers are required to orient an orbit in space. The six numbers, called orbit elements, describing the geometry of the orbit are all in line 2 of the TLE. A seventh number, the epoch of the TLE, is the number 10304.20566066 found in line 1 of the TLE. The epoch is the exact time that the orbit elements in the second line were valid. Satellite orbit elements change over time so the elements are only accurate at the instant of the epoch of the TLE. The epoch is given in Greenwich Mean Time (GMT). To decode the epoch we read the first two digits as the last two digits of the epoch year. The "10" translates to 20"10" for this example. The rest of the epoch is the day of the year. The day of this example is the 304th day from 31 December which counts out to 31 October in non leap years. The decimal part of the day is .20566066 days which is 4 hours 56 minutes and 9.081 seconds into the day.
Now we will discuss the six numbers representing the orbit geometry. Listed in order from the second line of our example TLE we have:

62.2806 = inclination, i
61.3884 = longitude of the ascending node, Omega
7489727 = eccentricity, e
276.5130 = argument of perigee, w
11.4192 = mean anomaly, M
2.03412498 = mean motion, N

There are three numbers that describe the size and shape of the satellite orbit (its path), and three numbers which describe the orientation of the satellite orbit with reference to the surface of the earth. As seen in the diagram below, all closed orbits describe an ellipse in the orbit plane with the earth located at one of the foci of the ellipse.

In the diagram above, the parameter a is called the semi-major axis and the parameter b is the semi-minor axis. The shape of the ellipse is given by the eccentricity, e, which is a number that describes the amount of "flattening" of the orbit shape. The eccentricity is defined by the relationship:

We see that the eccentricity is a dimensionless number between 0 and 1. The decimal point preceding the eccentricity in a TLE is omitted but it is understood that the eccentricity is less than 1. The eccentricity of our example satellite is actually 0.7489727. A perfect circle, where a = b, has an eccentricity of 0 and the foci are at the center of the the ellipse. As an ellipse flattens, a >> b, the closer the eccentricity is to 1 and the foci move out towards perigee and apogee.
Most satellite orbits have a low eccentricity. We can see in the diagram above that the size of the semi-major axis, a, would give a good idea of the size of a low eccentricity orbit. The semi-major axis does not appear explicitly in a TLE but it can be computed from the mean motion, N. The units of the mean motion in a TLE are revolutions per day. A geostationary orbit, where the satellite is stationary over a geographical location, would have a mean motion of 1.0027379093 days. This number is not exactly one due to the fact that a satellite must travel slightly more than one orbit per day to keep up with the earth's motion around the sun. So we have the conversion: one solar day (one revolution with respect to the sun) = 1.0027379093 sidereal days (one revolution with respect to the stars). Our example satellite makes 2.03412498 revolutions per day, which is slightly more than two revolutions per day in a very stretched out ellipse of high eccentricity. The size of the orbit can be calculated from the mean motion using the following equation:

Notice that there is an inverse relationship between the semi-major axis and the mean motion. As the mean motion increases, the semi-major axis decreases, and vice versa. The units of the semi-major axis, a, using this equation will be in earth radii. Thus, a geostationary satellite which makes one revolution per day, N = 1.002738, is at 6.6107 earth radii from the center of the earth.
The fifth number in the TLE describes the position of the satellite in the orbit plane. This is given indirectly by the mean anomaly, M. We see in our diagram that the position of the satellite in the orbit plane is given by an angle, theta, measured from the perigee position to the satellite. This angle is called the true anomaly. The relationship between the true anomaly and the mean anomaly, M, involves an intermediate quantity called the eccentric anomaly, E.

The mean anomaly is given in degrees in the TLE, but E and M will need to be in radians if used in the second equation. The second equation is famous in astrodynamics and is called Kepler's equation. Kepler's equation cannot be solved for E in closed form, so to work backwards from M to solve for the true anomaly requires numerical methods. In any use of these equations it is necessary to keep the quadrant ambiguities clear by noting that theta, E, and M are all in the same half plane of the orbit as divided by a line between perigee and apogee.
For a circular orbit, theta, E, and M are all equal. Mean anomaly gives the angular position of the satellite as measured from perigee as if the satellite were moving with uniform speed along the orbit path. Since satellites move faster in their orbits through perigee than they do near apogee the mean anomaly agrees exactly with the true anomaly only at perigee and apogee. Although all real orbits have non zero eccentricity, the eccentricity for most satellites is close enough to zero that the mean anomaly can give one a good approximate idea of the position of the satellite along the it's orbit path.
The plane of a satellite's orbit is oriented with respect to the earth by the remaining three elements in the TLE. The coordinate system geometry of the earth's reference system is the plane of the earth's equator, called the celestial equator, and a direction vector within that plane defining the x-axis, which is the direction of zero degrees rotation in the plane. Positive rotation angles are counter clockwise when viewing the celestial plane from above, on the side containing the northern hemisphere. The x-axis direction is chosen to be the vernal equinox which is also called the first point of Aries. It is located at zero degrees right ascension in the right ascension declination celestial coordinate system.

The plane of a satellite's orbit intersects the celestial reference plane defining a line called the line of nodes. At one node the satellite will pass from the northern hemisphere to the southern hemisphere. The ascending node is where the satellite passes from the southern hemisphere to the northern hemisphere. The angle from Aries to the ascending node is given by Omega in degrees.
The angle of tilt between the two planes is measured at the ascending node and is called the inclination, i. Inclination is usually less than 90 degrees but some satellites have retrograde orbits with inclinations greater than 90 degrees.
The last TLE orbital element left to discuss is the argument of perigee, w, or just called perigee for short. This angle, in degrees, is measured from the ascending node to perigee. Geostationary satellites strive to have an inclination of zero degrees and an eccentricity of zero to keep them right over the same spot on the equator. Either of these conditions would give our TLE a problem as it would be impossible to measure Omega if the inclination were zero and there would be no perigee if the eccentricity were zero. It is fortunate for the TLE convention that all real satellites have non zero inclinations and eccentricities.
For further study see:

- wikipedia:Orbital elements
- NCSU Physics page
- NASA Orbital Elements
- Celestrak Orbital Elements data sets
- SATOBS.ORG TLE description