Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time? 
PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely?
 A: Gravity is mediated by a spin two particle. Electromagnetism by spin 1.
Here is a link that answers your question:

even and odd spin do differ in that they require a product of charges with different signs to get attraction or repulsion:
spin even:

*

*$q_1 q_2 > 0$: attractive

*$q_1 q_2 < 0$: repulsive

spin odd:

*

*$q_1 q_2 < 0$: attractive

*$q_1 q_2 > 0$: repulsive

In the case of gravity, mediated by spin 2 particles, charge is mass, which is always positive.  Thus, $q_1 q_2$ is always greater than zero, and gravity is always attractive.  For spin 0 force mediators, however, there is no restriction on the charges and you can very well have repulsive forces.  A better rephrasing of the question is: "Why do particles of odd spin generate repulsive forces between like charges, while particles of even spin generate attractive forces between like charges?"

Goes on to derive this
A: If gravity is entropic, as suggested recently by Verlinde and earlier by others, gravity might be expected to be mostly attractive.
My speculative imaginary view of this has been that if the evolution of quantum physical objects is not perfectly conservative, then there might be processes which convert energy between scales. One possibility is that fields at the scales that determine gravitational interactions would be converted to lower scale matter that is currently not detectable. The result would be both an inflow and an outflow from regions that contain large masses, but on different scales and with different effects, attractive for matter that interacts more with the inflow, repulsive for matter that interacts more with the outflow.
At the human scale I suppose we have to imagine that we are pressed down by a flow of something that isn't the same as matter, it's the gravitational field if you will, and of course not an aether, metaphorically being sucked into the earth as food for a fundamental entropic process. Whatever exhaust there is from this process interacts with us enough less to be essentially undetectable on anywhere close to human scales, but would be detectable on either very large or very small scales.
Like I say, speculative, and also only a very small part of a whole entropic process. I can take a few or even a lot of downvotes, but pretend for a moment we're shooting the breeze at coffee. I haven't followed the Verlinde and other entropic gravity literature at all closely, so I don't know whether something like this of model has been suggested in a mathematical form, which would be required for it to be publishable (although being published definitely wouldn't be enough to make this not speculative).
If you ask for explanations for the established mathematics of General Relativity, I think the only currently possible response is speculation. GR is more grounded in empirical principles than in models that could be taken to be explanatory (which I say without prejudice insofar as I take empirically supported principled theories to be preferable to more-or-less ad-hoc models, except for the elusive question of how to imagine effective new empirically supported principles). I note that lurscher doesn't address your request for explanation.
A: It is not true at all that gravitation is always attractive. In fact, if it were always attractive, the universe would not be expanding at an accelerated rate right now and an inflationary period would not have occurred.
Currently the only known source of expansion components of gravity is the cosmological constant, which, incidentally, is precisely the physical quantity that our theories fail to predict by the largest amount: 120 or 60 orders of magnitude, depending on whom you ask.
A: When I was a schoolboy, our teacher of physics asked once one of our brilliant students (a girl), something like: "What is the nature of gravity?". She thought for a moment and answered: "I do not know. And what is it? Our teacher replied: "If I had known..."
As long as gravity is not derived from other nature features, it should be considered as a fundamental law. We may study its properties but we may not ask such questions - just by definition of a fundamental law. It is as an axiom - it is given as such.
A: In general relativity, in general but we can consider the most simple case of a spherically symmetric gravitational field, gravity is always attractive as long as you are at rest with respect to the gravitational field. However, if you are moving with respect to the gravitational field, gravity can sometimes be considered to be "repulsive". If you are falling in radially, from initially being at rest at high altitude, you initially accelerate towards the central mass. However, you will reach a point, if the central mass is compact enough like a black hole, where you start to decelerate and when you are infinitely close to the "Schwarzschild radius" you will move infinitely slow. This is all if your movement is observed from a distant observer.
In the classical sense of the meaning, gravity will become "repulsive" as soon as you start to decelerate. The reason for this deceleration is that in contrast to "classical Newtonian mechanics", where the force only increases the momenta by increasing the velocity, and accelerating particles in an accelerator where the electromagnetic force increase the momenta by increasing the "lorentz factor" setting the speed of light to be the speed limit, in general relativity you have the third effect that the speed of light (again as measured by a distant observer) around a spherically symmetric mass distribution decreases with the radial distance.
The gravitational force will always be "attractive" if you are moving in towards a black hole in the sense that your velocity as a fraction of the local speed of light constantly increases but your velocity, as observed by a distant observer, will if you get close enought to a black hole inevitably start do decrease and it that sense gravitation will become "repulsive".

Also, in practice, Nasa/JPL is using the force on the right side of this expression to mimic relativistic effects in the weak fields of our solar system:
$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}+\frac{v^2}{c^2}\right)\hat{r} +\frac{4GM}{r^2}\left(\hat{r}\cdot \hat{v}\right)\frac{v^2}{c^2}\hat{v}$$
You can see that this becomes repulsive, even if you are not moving with respect to the central mass at a radial distance of $r=4GM/c^2$ which is twice the Schwarzschild radius in Schwarzschild coordinates.
If your moving radially inwards this becomes repulsive, if I am not doing any mistake, at: $$v=\frac{c}{\sqrt{3}}\sqrt{1-\frac{4GM}{rc^2}}$$
Notice that JPL is only using the expression above in the weak fields of our solar system, it gives the right value of the so called "anomalous precession of perihelion" but it is not supposed to work in the strong field limit. The expression is provided as number 4-61 on page 4-42 in the official documentation, Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation.
A: The inverse square is apparently a consequence of conservation of momentum. For two particles in orbit, Newton showed that the orbit is planar, and Bertrand https://en.wikipedia.org/wiki/Bertrand%27s_theorem showed that the forces between the two have to be either of the inverse square k/r^2 or space spring/Hook's law. So Newton's law of gravity and that of Coulomb have conservation of momentum as the origin.
It is also worth noting that Hook's law can be shown to come out of/limiting case of the inverse square in the case of 'crowdedness', where there are too many interactants and very small space to move. This can be shown easily by taking three particles along a line interacting under the inverse square and give the middle a nudge keeping the end particles fixed. If we take it that there is no charge with zero mass, Coulomb's law follows too.  
Even without the help of Bertrand theorem, it is possible to derive the Maxwell equation from just charge conservation and its continuity equation. See this reg and quote: https://pdfs.semanticscholar.org/3251/31eadb62c8fdfdaaad7b21a308992ff3a4d2.pdf   '' We show how the covariant form of Maxwell’s equations
can be obtained from the continuity equation for the electric charge ''. Clearly the same can be done using mass conservation and we get the gravitomagnetic equations out of it.
In general, Maxwell/gravitomagnetic equations work for both attractive and repulsive forces. But if two masses are locked in an orbit, they must be under attraction. The facts that similar masses attract whereas similar electric charges repel has to rely on experimental knowledge.
New research from MIT on a new long-range attraction force connected to spin. https://www.youtube.com/watch?time_continue=10&v=1ZZcgBmS5W4 
A: Artwork:
dipole +- <--- some distance --->  +- dipole
Two dipoles are always attractive (or a dipole and another charge). 
If they are like this +- ... -+ or -+ ...+- the dipoles will rotate and the configuration become attractive +- ... +- or -+ ... -+ .  
They obey a 1/r³ relation.  
If you can consider that inside the baryons (neutron, proton) can exist a configuration of dipoles you have an answer. (read the book of Douglas Pinnow, 'Our Resonant Universe'. It is a monography of a model of particles where this happens).  
How do we go from 1/r³ to 1/r²? Integrate along the path.
Why? Explore the concept of polarizable vacuum.
The consensus is that gravitation is not electromagnetism, but in that way it is always attractive. And I like it.  
But the order of magnitude of gravitation is $10^{-35}$ (more or less, by memory) of EM, so how can it be EM?
Can you figure out two EM radiators in each dipole in opposition of phase (one the +, other the -) extremely near one of the other? Yes, the radiated EM field has to be extremely faint. 
(The connection with EM is much more compelling, IMO, that a connection with thermodynamics or other...)
A: Gravity is not always attractive:
Gravity is thought of as a weak boring force. But if you get fast spinning black holes and strong gravitational waves interacting, then you can get 'repulsion and attraction' effects happening out of purely gravitational forces.
So repulsive gravity does not exist in quiet spacetimes, but if you look at what happens when you shine a gravitational wave onto a rapidly spinning black hole, you will find that adjusting the frequency only slightly from the ideal level allows you to push the hole away from you (when the bh absorbs a wave) or pull it closer (when the bh adds energy to the radiation beam via superradiance). 
See:
http://arxiv.org/pdf/1312.4529v2.pdf - figure 4. With a Gravitational Wave 
