A 7th add 13 chord is often voiced r-5-7-3-13, sometimes leaving out the fifth. To pin this to a key, a G7(13)—G⁷⁽¹³⁾ if your chart can handle Unicode—is usually voiced G-D-F-B-E and is a common (in jazz and standards anyway) way to make the resolution from G7 to C more subtle and harmonically interesting. In G7 – C, the F resolving to E is hard to ignore, but in G7(13) – C, the E is already present, making the removal of the F more subtle. The dissonance between F and E (a major seventh interval) gives the G7(13) a richer sound.
The consensus on the web seems to be that this is actually a 13th chord, even though there’s no 9th or 11th present:
“In modern pop/jazz harmony … a thirteenth chord does not imply the quality of the ninth or eleventh scale degrees.” [Thirteenth on Wikipedia]
I disagree. Since the true 13th chord is an extension of the 11th chord, it should sound more like an 11th than a 7th. Elevenths almost always imply a suspended, missing, or nearly inaudible 3rd, but the 3rd in 7(13) is usually very prominent. If you really think G13 and G7(13) are the same chord, play these two voicings one after one another on guitar:
Clearly there is movement; they are different chords. If you see a 13th chord in a pop music chart, know that it is really a 7(13).
Some easy V7(13) to I resolutions on guitar (with the 3rd kept on top)
About the Eleventh (and its third)
As I mentioned, 11th chords almost never have the 3rd present*. You will almost never hear a B in a G11 chord. Some people write these as F/G, and, for voicing purposes this OK, but I dislike the notion that G11 should be thought of as an F chord. 1) it has a D in it. We could write it as F6/G or Dm7/G, but that’s taking us down the wrong path. 2) “G11” gives the reader a better impression that the chord will resolve to (and that our key is) major C.
*So it might really be a 9sus4, but writing it is as 11 seems a less egregious error than writing 7(13) as 13.