When space bends, what are the lines that are being bent? In an electric field diagram, the lines represent the electrostatic force vector at the position.    These lines are bent when you place a charge into the system.
What is the equivalent description of the "lines" that are being bent when a mass is placed into an area of space?
 A: 
When space bends, what are the lines that are being bent?

The straight ones.
For example: say you have two parallel laser beams traveling through space. Photons travel in a "straight line" more or less by definition, i.e. the path of a laser beam is the straightest line humans can produce, so it is a benchmark of sorts. Mathematically two parallel beams should never cross. Now we introduce into the scenario a massive object such as a star, past which the two beams pass on either side. The two beams are traveling in a straight line each before they pass the star; they travel in a straight line each after passing the star; and as far as we can tell, they travel in a straight line each as they are passing the star. So in theory, if they each travel in a straight line through their entire path, they should remain parallel after passing the star. However, because space is curved near the star, the straight lines that the lasers are traveling become unparallel, and somewhere on the far side of the star they cross paths.
Another example: if a planet is flying through space it should travel in a straight line if there's nothing pushing it some other direction (like a rocket engine or some such). Just for kicks we'll say it's traveling directly toward the nose of Ursa Major. Now let's say a star comes ambling along near the planet; due to the curvature of space around the massive star, the path of the planet deflects away from its straight Ursa-bound course, possibly even being captured in orbit around the star. There still isn't anything pushing the planet off course, but its "straight line" is changing due to the curvature of space.
Note that in this second case, the star will also deflect past (or orbit around) the planet by a small amount, the exact amount being proportional to the ratio of masses between the planet and the star. This is because the planet curves space a little bit, just as the star curves it by a lot.
A: The field line mind picture in General Relativity is probably not useful, at least not for me, aside from in very special cases, because


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*The dimension of the pictures you're trying to see at is too high for our everyday spatial intuition to help us much - you're trying to visualize a rank 2, $4\times 4$ tensor with 10 independent components (the metric), not a force vector as in electrostatics or Newtonian Gravity. In Newtonian or weak field Einstein equation gravity, the $g_{0\,0}$ component becomes the analogue of the gravitational potential $\phi$, so the mind picture you seek becomes that of Newtonian gravity with force lines along $\nabla\,\phi$. But these are not the lines of "bent space";

*Gravity is not a force in GTR. Nor can you define a global system of co-ordinates that allows you to define an "acceleration field" that would stand in for it. You can only define acceleration relative to locally comoving inertial frames, which is what an accelerometer measures. So, around a planet, for example, you could plot an acceleration field relative to the planet's surface. So you're really back to the Newtonian picture.
Don't get too hung up about the words "bent spacetime, warped spacetime", because you can't visualize them: our evolution in Neogene East Africa on Planet Earth did not kit us out for kenning such patterns - their recognition wasn't useful for finding food and water or keeping clear of Neogene Lions. They cause a great deal of needless feelings that "I'm never going to be able to understand this stuff". I really do think abstraction is your friend here: "bent spacetime" simply means that if you take a vector and parallel transport it around a loop in that spacetime, it will have changed to a different vector when you come back to your beginning point. Or, alternatively, bent spacetime means that geometry done there will not fulfil all of Euclid's axioms. Angles inside triangles don't add up to $180^\circ$. That's all there is to it. This variation around a path is called holonomy  - look this up here. The object used to measure this is a rank four object: it must take as inputs two vectors (which define the edges infinitessimal parallelogram that you imagine a vector being transported around) and spits out a matrix which defines the transformation any vector undergoes in being transported around a loop. Thankfully, we can pare this down to a rank two object which has simpler geometrical meanings, and one of the best elementary accounts of the meanings of all of this is to be found in Chapter 42 of Volume 2 of the Feynman Lectures.
A: If you plot grid-lines on a graph what are the grid-lines?
Well, on a typical 2D plot with x increasing to the right and y increasing upward, the horizontal grid-lines are the loci of constant y and the vertical grid-lines are the loci of constant x.
In 3D the grid lines are the loci of constant x and y for the ones that run parallel to the z-axis and of constant y and z for grid-lines running parallel to the x-axis and so on.
That is what is shown in those visualizations of bent space: loci of constant values for some set of time and space coordinates.
A: When people talk about curvature of space, they are usually talking about two related ideas: 1) the curvature of world-lines in 3+1-space-time or 2) the curvature of light paths in 3-space. These ideas are related as light behaves as the high-speed, low-mass limit.
1) Consider a 2D surface embedded in a 3D flat space. At any point on the surface, there is a tangent plane. If the surface is curved, then as we slide the tangent plane from one point to another, the angles it makes with respect to the fixed 3-dimensional axes changes.
Now, consider a particle moving at a constant unit speed along this surface. We can project its velocity vector onto the fixed 3-dimensional axes. At each point along the curve, the components of the velocity change, but the total magnitude of the velocity remains the same.
To make the connection with spacetime, we let the one dimension be time, the other two are space, and our constant unit speed is the rate of proper time of the particle. The curve traced out by the particle is its worldline, and the ratio of its spatial components to its time component is the observed velocity of the particle. If we consider a particle moving in a straight line in flat space, then a force causing it to speed up causes its world line to curve toward the spatial plane, and a force causing it to slow down causes the world line to curve toward the time axis.
2) To describe a 3D space which is not flat, I like to think about sponges. A flat space is represented by a homogeneous sponge. If we place a mass in a homogeneous sponge, then it pulls the sponge toward it with a distance-dependent force. The effect is that the sponge gets more and more dense as you approach the mass.
A beam of light seeks out the straightest, maximum spatial component, constant speed path it can find. While particles could slow-down/speed up when the density of the space-sponge changes, the beam of light does not have this option, so it deflects to optimize its path. 
