Using chord tensions in your melodies and chord voicings will make ordinary chords sound much more lush and "jazzy". They create more options of different colors in the sound of a chord.
The word "tensions" is really short for "extensions" because they are an extension of the basic seventh chord. Chords are basically a scale stacked every other note. For example, a C major scale is CDEFGABC with the notes numbered 12345678 and a C major7 chord is CEGB with the corresponding scale degrees 1357. The tensions of a C major7 are DFA with the numbers 246. Since the tensions are actually considered an extension of the seventh chord, we give them the numbers 9, 11 and 13.
9 is the same as the 2nd note of the scale
11 is the same as the 4th note of the scale
13 is the same as the 6th note of the scale
Often it's sounds nice to play the tensions above the chord tones but you also can place them between the chord tones as well. Some chord tensions work on some types of chords better than others, here's a breakdown of what to use and when:
On a major 7 chord, you can use 9's and 13's to your own taste. 11's clash against the major 3rd too much but you can use #11's, but usually only on major 7's that are NOT the Imaj7 chord (use them on IVmaj7, bIImaj7, bIIImaj7, bIVmaj7, bVImaj7 and bVIImaj7). It's almost become cliche to play a #11 on the final major7 chord of a jazz song.
On a minor 7 chord, you can use 9's and 11's and both sound awesome and really give a minor 7 a full, modern almost R&B sound. You can occasionally use a 13 on a minor chord but it will really stick out as it rubs against the flat 7 (which is sometimes a good thing).
On dominant 7 chords, you can play 9's and 13's. 11's are avoided because like the major 7 chord, it rubs against the major 3rd in a unpleasant way. In addition to 9 and 13, you can "alter" the tensions 9, 11 and 13 by flatting or sharping the 9, sharping the 11, and flatting the 13. This creates many options for different colors and are explained more fully in the Dominant V7 Chords section.
Finally, when you see a chord symbol such as C-11 or C13 it means that you should play all the available chord tones and tensions below as well. So C-11 has the 1,b3,5,b7,9 and 11. But if you see something like C-7(11), then you should just add the one tension listed. So C7(13) has the 1,3,5,b7 and 13. Sometimes you see chords like a C-6 but this means there is an added 6 but NO flat 7 it is written as a 6 instead of 13 because the number in a chord symbol only implies the chord tones below the number are played.