Rhythm

Webster's Definition: "the aspect of music comprising all the elements (as accent, meter, and tempo) that relate to forward movement"

My Definition: The flow or movement of a piece of music.
          ~It can be a little tough to really describe just what rhythm is - it  often seems to be something simply understood. You feel it and experience it when you listen to music. It often seems that it flows right through you from the music. When it comes to dancing, we say that some people have rhythm and other just don't. Rhythm is just something that's there in music. It simply exists.

It is, however, also a technical term used to describe an element of music. Rhythm is made up of three sub-elements: beat, meter and tempo.

Beat
Beat is really the big thing when it comes to rhythm. Beat is what any listener, no matter how musically knowledgeable, can usually find and follow along. It is the steady pulse of a piece of music.
Oftentimes when we think about beat, we think of drums. Drums and other percussion instruments are generally the main purveyors of the beat in a piece of music - at least in music from the last century or so. In fact, it can be quite difficult sometimes to find the beat in a lot of classical music, simply because of the way that they are composed and the instruments at their disposal. 
We, too, have steady beats: our heartbeats. And just like the way that your beats in a pattern of DA-dum-DA-dum-DA-dum, beats in music are also both strong and weak.
Meter
Meter is simply the way in which those strong and weak beats I just mentioned are organized in a piece of music. Generally, meter is relatively simple. There are three types of simple meters: duple, triple, and quadruple...
  • Duple Meter: this type of meter has a pattern exactly like your heartbeat - it goes DA-dum-DA-dum-DA-dum-DA-dum. In counting this meter out, it would go something like "ONE-two-ONE-two-ONe-two" and so on.
  • Triple Meter: this type of meter has one more "dum" sound than duple meter. If your heart was in triple meter, it would go DA-dum-dum-DA-dum-dum-DA-dum-dum. If you counted it out, you would say "ONE-two-three-ONE-two-three-ONE-two-three."
  • Quadruple Meter: this type of meter is like a double heartbeat - it goes DA-dum-dum-dum-DA-dum-dum-dum-DA-dum-dum-dum. You would count it out like "ONE-two-three-four-ONE-two-three-four."
NOTE: the technical terms for the "DA" and the "dum" are pick-up and downbeat, respectively. The pickup is strong (DA) and the downbeat is weak (dum).
Now, there are also some more complicated types of meter. I won't get into them too much, but they are complex meter, compound meter, polymeter, and mixed meter.
Tempo
The tempo of a piece of music is simply it's speed. It is how fast the beat goes. Whether it is your heartbeat or a musical beat, tempo is measured in BPM. For your heart, this stands for Beats Per Minute. For music, however, it stands for Beats Per Measure.

A great example for rhythm is the White Stripes' "Seven Nation Army." It has a very easy to find and follow beat, is in a simple meter, and has a rather moderate, sometimes fast, tempo.