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 $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$?
Best Answer
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).