When is a Banach space a Hilbert space? Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
Question: Are there other (simple) characterizations for a Banach space to be a Hilbert space?
 A: From the point of view of manifolds and curvature the following result is valid:
A Banach space is a Hilbert space if and only if it is a NPC (non-positive curvature) space.
 http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-Theodor_Sturm/papers/paper41.pdf
A: S. Kwapien proved in "Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients" (Studia Mathematica 44, 1972) that the Banach spaces in which the vector valued Khintchine inequality is valid, are Hilbert spaces. To be more precise, a Banach space $X$ is isomorphic to a Hilbert space if and only if $X$ has type $2$ and cotype $2$, i.e., if there are constants $c,C>0$ such that for all $n\in\mathbb N$ and all $x_1,\dots,x_n\in X$ 
$$
c \Big(\sum_{i=1}^n\|x_i\|^2\Big)^{1/2} 
\leq \bigg(\int_0^1 \Big\| \sum_{i=1}^nr_i(t)x_i \Big\|^2 \bigg)^{1/2} 
\leq C \Big(\sum_{i=1}^n\|x_i\|^2\Big)^{1/2}
$$
A: From this article by O. N. Kosukhin:

A real Banach space $(X, \|\cdot\|)$ is a Hilbert space if and only if for any three points $A$, $B$, $C$ of this space not belonging to a line there are three altitudes in the triangle $ABC$ intersecting at one point.

Many other references show when Googling

"is a hilbert space if" banach

A: Bessaga and Pelczynski wrote a survey on Banach spaces. The chapter 4 is devoted to this question. 
http://matwbn.icm.edu.pl/ksiazki/or/or2/or214.pdf
A: More characterisations are in the book of Haim Brezis (Analyse fonctionnelle), at the appendix of Chapter 5. I will copy two of these below, toghether with the references:


*

*If $ \dim(E)\geq 2 $ and every subspace $ X\subset E $ of dimension $ 2 $ is the image 
of a bounded projector $ P $ such that $ \|P\| = 1 $, then $ E $ is isometric to 
a Hilbert space 
(Kakutani, Japanese Journal of Mathematics, 1939);

*if $ \dim(E)\geq 3 $ and the map $ T $, defined as the identity on the unit ball and 
as $ u/\|u\| $ when $ \|u\|\geq 1 $, is lipschitzian with constant $ 1 $, then 
$ E $ is isometric to a Hilbert space
(de Figueiredo; Karlovitz, Bulletin of the American Mathematical Society, 1967).


Also, if $ E $ is isomorphic to all its infinite-dimensional subspaces, then it is isomorphic to a separable Hilbert space (Gowers, Annals of Mathematics, 2002).
A: Just two isometric/isomorphic characterizations:
A Banach space $X$ is [isometric to] a Hilbert
space if and only if there exists a Banach space $Y$ and a symmetric
bilinear mapping $f:X\times X\rightarrow Y$ satisfying 
$||f(x,z)||$
$=$ $||x||\cdot||z|$| for all $x,z$ $\in$ $X$. 
[J. Becerra Guerrero & A. Rodriguez-Palacios]
A Banach space is [isomorphic to] a Hilbert space iff it is uniformly
homeomorphic to a Hilbert space. [Per Enflo]
A: Yes, there are many (simple) characterizations of when a normed space is an inner product space.  Here are two book references, one with Google preview (Inner Product Structures: Theory and Applications
By V.I. Istratescu), the other you can hopefully get at your library (Characterizations of Inner Product Spaces by Dan Amir).
A: In this simple note http://arxiv.org/abs/0907.1813 (to appear in Colloq. Math.), Rossi and I proved a characterization in terms of "inversion of Riesz representation theorem".
Here is the result: let $X$ be a normed space and recall Birkhoff-James ortogonality: $x\in X$ is orthogonal to $y\in X$ iff for all scalars $\lambda$, one has $||x||\leq||x+\lambda y||$.
Let $H$ be a Hilbert space and $x\rightarrow f_x$ be the Riesz representation. Observe that $x\in Ker(f_x)^\perp$, which can be required using Birkhoff-James orthogonality:
Theorem: Let $X$ be a normed (resp. Banach) space and $x\rightarrow f_x$ be an isometric mapping from $X$ to $X^*$ such that
1) $f_x(y)=\overline{f_y(x)}$
2) $x\in Ker(f_x)^\perp$ (in the sense of Birkhoff and James)
Then $X$ is a pre-Hilbert (resp. Hilbert) space and the mapping $x\rightarrow f_x$ is the Riesz representation.
