In the simplest of terms, the wobble effect is a method of detecting the existence of planets around stars more distant than the Sun, by observing the gravitational influence the planets around it have on the star. The effect of gravity on all objects in space is mutually consistent. Just as the gravity of a large body like a star draws smaller objects towards the star, The star itself is drawn equally towards the object. From an Earthly perspective, this causes the star to move back and forth, or give the appearance of wobbling, as planet or planets revolve around it.
To date, no one has ever detected the existence of planets beyond our own solar system with an optical or radio telescope. The reason is uncomplicated. Stars are usually hundreds if not thousands or millions of times more massive and luminescent (bright) than planets orbiting them. The one exception to this would be binary or tertiary stars, where two or more stars are in orbit of each other. It is not likely that star clusters like this would support planetary formation at all, due to the immense gravitational tides which would be present. The problem with seeing planets is that the stars they obit drown out any reflected light or electromagnetic radiation coming from the planets. It is a perplexing situation for astronomers, who know that planetary systems exist, but can not prove it through discrete observations.
The best hope for one day actually seeing a planet around some distant star will likely come from the Hubble Space Telescope (HST), or some other instrument or experiment in space. For instance, if the path of some far off asteroid of just the right size happened to eclipse a star while HST was pointed at it, for just that brief moment, Hubble might catch a glimpse of what would probably be a large planet at a point in its orbit where a maximum of the stars light was being reflected off it. On the other hand, it is more probable HST will be hit by a marauding asteroid and rendered nothing but space debris. For now, studying the wobble of stars is the best way to discover planets.
Detecting the wobble of a star is a painstaking process which requires making regular observations of the star for months and years. Making these observations requires some very sensitive and specialized equipment, not something your going to do with a backyard amateur telescope. The stars movements are so minute, and depending on the planetary orbital frequency, observing the wobble effect may require a very long time. If you visit the following link at NASA's JPL web site, there is an interesting article on the discovery of the first planet using the wobble method. You will find a 30 second video clip there you can download and view, which shows the wobble effect on a star. http://planetquest.jpl.nasa.gov/news/firstFind-images.cfm
If some aliens were viewing our solar system from some far off quadrant of the galaxy, The most obvious wobble effect they would notice would be caused by the planet Jupiter. The gravitational influence of Jupiter on the Sun is much greater than that of all the other planets combined, and thus results in the greatest degree of wobble exhibited by our local star. But, since Jupiter's average orbital distance from the Sun is about 3/4 billion miles out there, and it takes about 12 earth years to complete a single orbit, it might take quite a while for some alien life form to be able to witness Jupiter's effect on Sun wobble.
On the other hand, Mercury, the closest planet to the Sun, completes an orbit every 88 Earth days. For it's size, Mercury is quite dense, having about two thirds of the gravity of Earth, while being only slightly larger than earths moon in size. It too tugs on the Sun and causes a wobble. In effect, there are a whole bunch of wobbles associated with our Sun, including some anomalous perturbations that occur when planets lining up on one side of the Sun exaggerate the wobble in one direction and at other times dampen it. By understanding the nuances of star wobble over the past few years, astrophysicists have been able to predict the existence of planets and even have a pretty good idea of their proximity, orbital velocities and other planetary attributes.
The discovery of star wobble has provided humans with the first confirmation that there are other planets circling around distant stars, some of them similar to Earth and others very different. Perhaps someday we will get the opportunity to see one of these planets for the first time, but for now we can detect there existence and study them by the gravitational effect they have on their star, through the wobble method.