Elliptic curves with Mordell-Weil group Z/2Z over Q This question is not very precise; I hope it is suitable for the site.
I have come to a situation where I have to study rational points on an elliptic curve defined over $\mathbb{Q}$. I don't know much about the curve, let alone its equation. I already have one rational point, which sits on a bounded real connected component. What I want to avoid is that this is the only rational point (other than the marked point).
I am not sure what to use about my curve that will help me get there, so I turn the questio the other way round:

What is known about elliptic curves $E$ over $\mathbb{Q}$ such that $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$?

 A: I think not so much is known.
Conjecturally, elliptic curves (defined over $\mathbb{Q}$) of rank $0$ have density $1/2$, and since at any rate only a finite number of cases for the torsion part of $E(\mathbb{Q})$ are possible (by Mazur's celebrated result), I guess that elliptic curves with $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$ should have density $1/2$ too. 
For this reason, it seems to me that a complete classification is out of reach.
If you want to see an infinite family of cubic curves with this property,  take
$y^2=x^3 + px$
where $p$ is a prime number such that $p \equiv 7$ (mod $16$), see [Silverman-Tate, Rational points on elliptic curves, p. 105].  
A: Mazur's theorem ensures that there are exactly 15   possible cases for the torsion part of the Mordell-Weil group of an elliptic curve: the cyclic groups $\mathbb{Z}_n$ (with  $1\leq n\leq 10$ or $n=12$) and the groups $\mathbb{Z}_2\times\mathbb{Z}_n$   for $n=2,4,6,8$.
In his paper Universal Bounds on The Torsion of Elliptic Curves, Proc. London. Math. Soc.(1976) 33, 193-237 , Daniel Sion Kubert (who was a student of Mazur) presents in table  3 (page 217) a list of parametrizations for the different possible cases. 
In particular,  curves with a  $\mathbb{Z}_2$ torsion are parametrized by the following family:
$$
\mathbb{Z}_2\ \text{torsion}:\quad  y^2=x(x^2+a x+ b), \quad  b^2(a^2-4b)\neq 0.
$$
The example given in Francesco's answer is a special case with $a=0$. 
As another example, the case with torsion $\mathbb{Z}_2\times \mathbb{Z}_2$ is parametrized by the  Legendre family:
$$
\mathbb{Z}_2\times\mathbb{Z}_2 \  \text{torsion}:\quad   y^2=x(x+r)(x+s), \quad r\neq 0 \neq s \neq r.
$$
A slight generalization of the Hesse family parametrizes the curves with torsion $\mathbb{Z}_3$:
$$
\mathbb{Z}_3 \  \text{torsion}:\quad   y^2+a_1 x y +a_3 y =x^3, \quad a_3^3( a_1^3-27 a_3)\neq 0.
$$
For the other groups you might have to use Tate's normal form 
$$
E(b,c): \quad y^2+(1-c)x y - b y =x^3- b x^2
$$
and the condition for a given torsion is expressed as an algebraic condition on $b$ and $c$.
For example for $\mathbb{Z}_4$, we have $c=0$ and $b^4(1+16b)\neq 0$, which gives: 
$$
\mathbb{Z}_4 \  \text{torsion}:\quad   E(b,c=0): \quad y^2+x y - b y =x^3- b x^2, \quad b^4(1+16b)\neq 0.
$$
For a review, you can  read chapter 4 of the  book of Husemoller . A friendly short review is also available in section 2 of this string theory paper  by Aspinwall and Morrison ( they don't present all the 15 cases but for those they analyze, they express everything in Weierstrass form).
