Root system : Constructing the Dynkin diagram 
this is the image of the root system of $A_2$ i found online can someone explain to me how we construct the Dynkin diagram from this and why it's different from $D_2$

Also, why the root systems are represented like that what is the geometric meaning?
 A: A system of roots is used to encode the structure of certain Lie groups/certain Lie algebras. These have an associate group (the Weyl group) whose structure can be encoded by a root system, and the data of a root system is equivalent to the data of the associated Dynkin diagram.
The Weyl group is a finite reflexion group generated by orthogonal reflexions with respect to hyperplanes. These hyperplanes are the ones orthogonal to the roots. So, for instance, the Weyl group associated to your first diagram is the group generated by the 3 symmetries fixing the equilateral triangle, which is the dihedral group $D_6$ (with 3 symmetries and 3 rotations), also isomorphic to the symmetric group on $3$ elements. For the second one, the Weyl group is isomorphic to $(\mathbb Z/2) \times (\mathbb Z/2)$, generated by two commuting reflexions in the two orthogonal directions (it is "reductible" into a product).
To get the associated Dykin diagram (which completely determines the root system), you need a basis of the root system (which will be the vertices of your diagram), and then the angle between two elements of the basis must be $2\pi/m$ with $m$ an integer (because of the geometry of the action of the Weyl group), and you write "m" on the edge between the corresponding vertices (with no edge if $m=2$ and no label if $m=3$). In each of your examples, $(\alpha, \beta)$ is such a basis, and I let you draw the corresponding diagrams.
If you want to read further (in particular, for the definition of a basis), there is a treatment of root systems with only basic group theory and linear algebra in Benson and Grove's book on finite reflection groups, which can be a good starting point. Also, I think you might be interested in Fulton and Harris' book on representation theory. Or any book of representation theory, really (but theirs is very good !)
