I have to add something to the explanation above.
First of all, it's not necessary that a dodecaphonic composition begins with a melody presenting the row. For instance, in Schoenberg's opus 33, it is presented in three chords.
Other thing you must pay attention to is the repetition of tones: it happens often. Sometimes the series appears at various voices simultaneously (Schoenberg's opus 37: in the very beggining, 2nd violin, viola and cello share the series); sometimes, the composer uses more sophisticated processes such as troping, permutation, hexachords, elision, when the row is manipulated, reordered, overlapped.
By the way, one shall know the meaning of the terms atonality, dodecaphony, and serialism. They're not the same. Atonal is a composition that is not tonal. Period. Of course, dodecaphonic and serial music are atonal. However, when someone talks about dodecaphony it implies the use of a series of 12 notes (as explained above), but there are series of less than 12 notes, hence not
dodecaphonic. In the case of serialism, the term have a broad meaning: it refers either to orthodox serial music (Second Vienese School) or to other uses like Total Serialism (Boulez, Stockhausen).
Mr. Minor9th refers to people who think this technique is "cold" or "mathematic", and this is a common mistake. Every composer that did use serialism done it in a different way, and the musical results are as different as (or even more) the ones between Bach, Handel, Telemann and Rameau (just to quote a few late Baroque masters, whose language is sometimes also refered as "cold" or "brainy"). It's important to say that many composers don't follow strictly the principles, bending the "rules" according to their intentions and styles.
Last but not least! Mr. Pies, if you (or someone else) are interested in the serial method, a very good idea would be the reading of some textbooks. Here is a starting background:
FORTE, Allen. The Structure of Atonal Music.
RAHN, John. Basic Atonal Theory.
LESTER, Joel. Analytic Approaches to Twentieth-Century Music.
STRAUS, Joseph N. Introduction to Post-Tonal Theory.
PERSICHETTI, Vincent. Harmony: creative aspects and practice.
DALLIN, Leon. Techniques of Twentieth Century Composition: a guide to the materials of modern music.
SALZMAN, Eric. Twentieth-Century Music: an introduction.
ANTOKOLETZ, Elliott. Twentieth-Century Music.
GRIFFITHS, Paul. Modern Music and After: directions since 1945.
MORGAN, Robert P. Anthology of Twentieth-Century Music.
Further, there are several articles and more specific books dealing with particular composer or method variants. Of course, the reading of scores is fundamental. In the Morgan anthology you have some examples, and there are other books like this. Anyway, a list of basics:
Schoenberg:
Klavierstucke opus 23;
Suite opus 25; Variationem fur Orchestra; Third and Fourth String Quartets; Moses und Aron;
Klavierstucke opus 33a and 33b; Violin concerto.
Webern: String Trio opus 20; Symphony opus 21; Concerto opus 24;
Variations opus 27 (piano); String Quartet opus 28; Cantata n.1 opus 29; Variations opus 30 (orchestra).
Berg: Lyrische Suite; Lulu; Violin Concerto; Der Wein;
Messiaen: Modes des Valeurs et Intensites
Boulez:
Structures; Le Marteau sans Maitre;
Piano Sonatas (in particular, the second).
Stockhausen: Kreuzpiel; Gruppen; Gesang der Junglinge; Punkte; Kontrapunkte.
Babbitt:
Semi-simple variations;
Three compositions for piano; String Quartets (specially n. 3);
Partitions; Tranfigured Notes.
Stravinsky:
Movements; Cantata; In memoriam Dylan Thomas.
Other works by: Krenek, Dallapiccola, Berio, Nono, Barraque, Bennett, Roslavets (a pioneer), Maxwell Davies, Ferneyhough, Gerhard...there are hundreds or thousands more...

Any further help, please let me know. Best wishes!