Locally refining mesh over cylinder in Gmsh
 

I want to mesh a cylinder in Gmsh with a finer mesh at a particular circular location along its height. I have succeeded partially with the code below:

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SetFactory("OpenCASCADE");

//Create a circle
Circle(1) = {0, 0, 6, 1, 0, 2*Pi};
Curve Loop(1) = {1};
Plane Surface(1) = {1};

//Extrude into a cylinder 4 units along Z-direction and 6 units along -Z direction
Extrude {0, 0, 4} {
Surface{1};
}

Extrude {0, 0, -6} {
Surface{1};
}

//Convert Circle 1 into transfinite line so that a fine mesh can be obtained in its vicinity
Transfinite Line{1}=100;

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The mesh is coming out fine (when I do a 2D or 3D mesh) except at the location where the generator of the extruded cylinder intersects the circle (here the mesh is coarse). Any ideas on how to overcome the problem and obtain a uniformly fine mesh at Circle(1)?

At the moment, you are only defining the cell size along the line. You should also define the cell size elsewhere. For instance, define the cell sizes at the points:
Characteristic Length {1} = 0.1;
Characteristic Length {2, 3} = 1;

Your solution worked :) Thanks a lot.

But what do points 1,2,3 correspond to? Are they the three points at which cylinder's generator-line intersects the three circles?

You are welcome. If you visually look at the geometry, it contains only 3 points. The points are the start/end points of each circle.