As a follow-up to Richard's answer, I'd like to give an example of the way a row or series is used in a composition for several instruments, and explain how the series to be used for a composition was chosen.
Here is a simplified version (omitting dynamics and articulation) of the opening measures of Schoenberg's String Quartet No.4 op.37:
X:1
L:1/8
M:4/4
K:C
%%score V1 V2 VA CL
V:V1 clef=treble name="Vn.I"
V:V2 clef=treble name="Vn.II"
V:VA clef=alto name="Viola"
V:CL clef=bass name="Cello"
% 1
[V:V1] "P0""_0"=D4 "_1"^C4 | "_2"=A,A,A,"_3"_B, "_4"=F2 "_5"_E2 | "_6"=EEE"_7"=c "_8"_B4 | ("_9"=G4G)"_10"^FF"_11"=B |(=B8 | B3) z "I5"z4 | "_0"=gz/2g/2 g z z2 "_1"_a z | "_4"=e z z2 z2 "_6"=fz/2f/2 | f z z2 z4 |
[V:V2] z2 "_3"_B, z "_8"_A, z "_10"^F z | z2 "_6"=E z "_11"=B, z "_1"^C z | z2 "_9"=G z "_0"=D z "_5"_E z | z4 "_1"^C2 C2 |^C2 "_5"^D2 (D4 | D) z "_6"=E z z "_0"=G"_1"_A"_2"=c | ("_3"=B4 B)("_4"=e"_5"^F2) | z "_6"=F"_7"=A"_8"^c ("_9"=D"_10"_e') (e'2 | (e'2) "_11"_B6) |
[V:VA] z2 "_5"^D, z "_6"=E, z "_9"=G, z | z2 "_7"=C, z "_10"^F, z "_2"=A, z | z2 "_11"=B, z "_2"=A, z "_4"=F, z | z4 "_0"=D,2 D,2 |=D,2 "_4"=F,2 (F,4 | F,) z "_8"_A, z z4 | "_2"=C,z/2C,/2 C, z z2 =c z | "_3"=B z z2 z2 "_8"^C,z/2C,/2 | C, z z2 "_9"=Cz/2C/2 C z |
[V:CL] z2 "_4"=F,, z "_7"=C, z "_11"=B,, z | z2 "_8"_A,, z "_9"=G,, z "_0"=D,, z | z2 "_10"^F, z "_1"^C, z "_3"_B,, z | z4 "_2"=A,,2 A,,2 |=A,,2 "_3"_B,,2 (B,,4 | B,,) z "_7"=C, z z4 | [K:C alto 4] "_1"_Az/2A/2 A z z2 "_0"=G, z | "_5"^F z z2 z2 "_7"=Az/2A/2 | A z z2 [K:C bass] "_10"_E,,z/2E,,/2 E,, z |
From the start up to the middle of measure 6, the first violin plays the non-transposed prime version of the row (often indicated as P0). After that, the second violin plays the inversion of the row transposed up by 5 steps (often indicated as I5).
X:1
L:1/1
K:C
%%score V1 V2
V:V1 clef=treble name="P0"
V:V2 clef=treble name="I5"
% 1
[V:V1] "_0"d "_1"^c "_2"A "_3"_B "_4"F "_5"_E "_6"E "_7"c "_8"_A "_9"G "_10"^F "_11"B
[V:V2] G _A c B E ^F F A ^c d _e _B
While P0 is being played by the first violin, the other three instruments do not each play a version of the row. Instead, they work together so that every measure (or group of measures 4, 5 and 6) contains all twelve pitch classes. If we number the pitch classes in the rows from 0 to 11 (as is conventional), these are the notes played by each instrument:
P0 | | |
V1 0 1 2 3 4 5 6 7 8 9 10 11
V2 3 8 10 6 11 1 9 0 5 1 5 6
VA 5 6 9 7 10 2 11 2 4 0 4 8
CL 4 7 11 8 9 0 10 1 3 2 3 7
You'll see that these zones don't strictly correspond to measures in the score; e.g. the 2 is not played in measure 1 but at the beginning of measure 2.
Starting from the middle of measure 5, the second violin plays I5, but now the other instruments work in a different way; they combine to form chords of adjacent notes 0-1-2, 3-4-5, 6-7-8 and 9-10-11. So the I5 series is played twice, once by the second violin, and once by the other three voices combined.
I5
V1 1 4 6
V2 0 1 2 3 4 5 6 7 8 9 10 11
VA 2 3 8 9
CL 0 5 7 10
As you can see, the idea of using the row or series isn't limited to the horizontal dimension, but is combined with the vertical in different ways.
This brings us to an important point: the rows used by Schoenberg are not chosen randomly. He uses rows that have interesting properties when combining different versions, and one of those properties, "hexachordal combinatoriality", can be seen in the example above. If you look at P0 and I5, you'll notice that the first 6 notes of P0 are the same as the last 6 notes of I5, and vice versa. This means e.g. that if two instruments play these two versions of the series, each group of six notes also contains all twelve notes.
Webern often used rows that were themselves inversions and transpositions of a shorter row of 3 or 4 notes. Again, this is done so that the combinations of different versions of the row would have interesting properties, e.g. that they contain the same adjacent notes (like the adjacent D-C#-A in P0 and A-C#-D in I5 in the example above).