If we consider the sun, then space-time is curve around it. My question is that what is the kind of curvature of space and time around the black hole. Is that space and time more curved around the black hole. Can anyone explain me about that?
Space-time, as you said curves around a body in space, and the greater the mass of the object the greater the curvature, this, in the simplest way can be described using light cones and Schwarzschild. Black-holes create many odd situations in space in which odd events take place. Black holes, although often though of as places in themselves, they have their own geography. There are two checkpoints to look for when one falls in a black hole these are called, the Horizon and the Singularity. The Singularity is the easiest to explain, in terms of description.
As you can see the singularity is the center point of the black hole, it also shows us the relative place of the Horizon, our second checkpoint. The Horizon is defined by the size of the black hole. The calculation in how to work out the Schwarzschild radius, (where to place the Horizon) is stated below:
The important thing about the Horizon is that NOTHING SPECIAL happens if one were to pass it, on e would feel no special sensation and may not even know they had passed the horizon unless they had a way of measuring it. The only interesting thing that happens at the Horizon is that when someone else watch one they would see an odd situation, but that would be going on to a completely different discussion point. The Singularity is the only place in which one would feel anything, and that thing would a infinite tidal forces, ripping one apart because gravity would be infinite. This is where we come back to your original question, gravity is NOT a force, it is only talked of as a force so that it is simple to imagine, but a curvature of space, and therefore curvature would be infinite. This is only at the singularity though, at the horizon the curvature would not be quite as intense. If you want to know the exact equation then look up Ricci/Riemann Curvature equations as I have chosen to describe this is in a non-mathematical sense. Below I have left a diagram of a black hole to help you visualize it.
I hope I helped and did not ramble too much. If you found my explanation helpful and not too boring then look up more and even try listening to Leonard Susskind, or Brian Greene, both really interesting and have videos on YouTube. It is really interesting so dive as deep as you can, then go even deeper!