What is the meaning of space-time curvature? What is the difference between the Space-time curvature and Space curvature? 
 A: Spacetime curvature is mathematically equivalent to the presence of so-called geodesical deviation of timelike geodesics. In other words there are freely falling bodies starting from points close to each other and with similar velocities which measure a nonvanishing relative acceleration. This is the most direct physical meaning of a nonvanishig Riemann curvature tensor in a spacetime.
Space curvature has a similar, but not identical, interpretation in the extended rest space of an observer (assuming that the metric induced on that space from the one in spacetime is stationary with respect to the notion of time adopted by the observer). There is a "relative acceleration" referred to the natural length parameter (instead of proper time) between geodesics. Here geodesics can be defined in terms of their variational definition, since the metric is positively defined. They are the shortest lines joining pairs of given points.
A: "Space curvature" refers to the geometry of a spatial-slice of space-time, with a constant time coordinate (so the slice has no time dimension). "Space curvature" is what the common rubber-sheet-analogy refers to, mimicking Flamm's paraboloid, which represents the geometry of a spatial slice through the Schwarzschild metric:
http://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid
"Space-time curvature" refers to the geometry of the 4D space-time, or a slice with a non-constant time coordinate (so the slice has a time dimension). "Space-time curvature" is rarely shown, but it looks something like this:
http://www.relativitet.se/spacetime1.html
Here another comparison of the two types of slices of spaces-time. One with time, and one purely spatial:
http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html
A: This is long, but I think some people might benefit from a more elementary answer. 
Sometimes, on a map of the world, you will see a figure that looks like this:

It is a scale, showing how much distance on the map corresponds to 1000 miles in the world. That strange funnel shape is because the scale changes the farther you get from the equator. If you are trying to estimate distances near 80° latitude, you use the horizontal line at the top. If you are interested in the mid-thirties, where I live, you would use the width of the figure between the 30 and 40 marks. A scale for a map showing just a small part of the earth is just a simple line with some tick marks, but when you flatten out the curved surface of a globe to to make a map, you have to stretch parts of it out, and maybe squeeze other parts, and so the scale can’t stay the same over the whole map.
You can also show this changing scale directly on the map. Suppose there is a small circle on the globe everywhere the lines of latitude and longitude cross. When you stretch the surface of the globe to flatten it out, the circles will get stretched too. If each circle represents a 100-mile radius on the real earth, then you will see how far 100 miles is on the map at that point. People don’t do this on ordinary maps because it adds too much clutter, but it does help you understand how each kind of map distorts the real features. For instance, this is the famous Mercator Projection (taken, like the other examples here, from Wikipedia; citations at the end):

You can see how the southern and northern parts got stretched more as the globe was flattened. This projection is the one that the scale diagram shown earlier is for. 
In indicating the changing scale, these circles also tell you that the surface has to be curved. Think about how this map would be deformed if all the circles changed to be the same size, tugging the rest of the map with them. The northern and southern parts would pinch together, and the whole map would start to curl. The circles along the equator would get a little fatter, making the waist bulge. If there were more circles in between, they would tug the surface by intermediate amounts depending on where they were. Once the circles were all the same size, if you lined everything up right you would have a globe. (Except that the areas near the poles would be missing.)
If this is too abstract, think of knitting a skullcap with stretchy yarn and then stretching it out flat. The little loops of yarn would be bigger near the edge where the cloth was stretched the most, and you could use that to predict something about the shape that the skullcap would relax into when you let go. To be fair, you cannot predict the exact shape - skullcaps are not rigid. But whatever the shape is, it will be curved. There are mathematical ways to indicate how curved a surface is at every point, and you can calculate what those values are just from the ways the scale changes as you move around the map, without even thinking about it in 3D.
It would be a mistake to think that the scale is always the same in all directions. That’s usually not true, and it’s one of the things that make the Mercator map special - all of the scale markers are circles. For instance, the Robinson projection adjusts the north-south and east-west scales differently in different places, so that land areas near the poles aren’t quite so large, but shapes aren’t too terribly distorted either. On this map, the scale markers are ovals at different angles, indicating the direction where stretching was greatest.

In general relativity, time slows down as you get closer to a massive object, and radial distance between concentric shells becomes less than what you would calculate from the circumferences of the shells. This is nothing other than a change in the scale of a map. You experience your life in a small patch of spacetime that is very flat. That means you can make a square with four right angles and all four sides equal, and clocks tick at the same rate no matter where they are in your room. When you think about space and time around a black hole, you naturally  think of it as if it were a space and time that follows those rules. That mental model is the “flat piece of paper” your map is drawn on. When people say that time passes more slowly close to a black hole, they are essentially drawing the tick marks for time farther apart in that part of the map. In the same way, the spatial tick marks towards and away from the center of the black hole (or other massive object, really) are closer together near the object. In the maps of the world, we used little ovals rather than tick marks, but we could have used little mini-scale markers with tick marks indicating a given number of miles in the north-south and east-west directions instead. They would have different spacings in different parts of the 2D map, just as the time and space tick marks are separated by different amounts in our 4D mental map.
So, don’t think the changes in the pace of time and in radial distance are just related to the curvature. They are the curvature. We have no reason to believe there is a higher dimensional space that spacetime is curved through in the way that the earth’s surface is curved in three-dimensional space, and we don’t need to care. We can infer and even measure that the correct map scale differs in different places, and that’s curvature.
Both maps are from Wikipedia and are by Eric Gaba (Sting - fr:Sting). 
First map: Data : U.S. NGDC World Coast Line (public domain), GFDL, https://commons.wikimedia.org/w/index.php?curid=4677929
Second map: [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 4.0-3.0-2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/4.0-3.0-2.5-2.0-1.0)], via Wikimedia Commons
The variable scale bar is public domain.
