When did people start thinking of elliptic curves as groups? I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to use the word group.  (Edit: Taking another look, at least some of Cassels' papers from this period do use the word group.)
Weil - L'arithmétique sur les courbes algébriques (footnote 1, p. 281) says:

Afin de réserver le mot de groupe au sens qu'il a pris depuis Galois, je parlerai toujours de systèmes de points, bien qu'on air l'habitude en géométrie algebrique de parler de groupes de points sur une courbe.

Question: When did it become common parlance to call elliptic curves groups?
 A: The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was 
Juel [Ueber die Parameterbestimmung von Punkten auf Curven zweiter
und dritter Ordnung. Eine geometrische Einleitung in die Theorie der
logarithmischen und elliptischen Funktionen, Math. Ann. 47 (1896), 72-104].
Poincare, in his important article referred to by ThiKu, does not use the concept of groups (this is the point of Schappacher's article). Even Mordell proved his theorem, namely that the group of rational points on an elliptic curve is finitely generated, without using the notion of a group: this was only done by Weil.
For a long time, most people interested in elliptic curves regarded them as
a variety of diophantine equations, and preferred thinking about secant and
tangent methods instead of group operations. Only when it became clear in 
the 1960s that the rank of an elliptic curve could be computed by applying
all kinds of homomorphisms and determining the orders of kernels and images of such homomorphisms, the group theoretic point of view became indispensible. 
A: H. POINCARÉ
Sur les propriétés arithmétiques des courbes algébriques
Journal de mathématiques pures et appliquées 5e série, tome 7 (1901), p. 161-234. 
http://sites.mathdoc.fr/JMPA/PDF/JMPA_1901_5_7_A7_0.pdf
