How do you integrate the meshes of cylinder and a plane, when you position the cylinder on top of the plane?
By using ctrl-j you can join the meshes. However the vertices are not adjusted to form the new object.
One way is this:
Remove the bottom face of the cylinder, and the face of the plane (Remove Faces only). Subdivide the edges of the plane, so that the no. of vertices roughly match that of the cylinder.
Select the “matching” vertices of the plane and the cylinder. Press Shift-F to “fill”; then Alt-F to beautify.
Another way - in this case.
Inset the face of the plane a bit. Subdivide. Again inset the inner face till it’s the size of the cylinder.
Subdivide the inner faces again (using “multiple cuts” so you get enough vertices to form the circle in the next step).
Delete enough inner vertices so that you have a hole of the size of the cylinder.
Select the outer vertices of the hole and press Ctl-Shift-S (“To sphere”). Type 1 to make this a perfect circle. Move the circle where you want your cylinder to be. Extrude it.
(This is actually easier to do than it sounds - or is to explain 
Good luck 
It kind of depends what you want to do with it. In some cases you don’t even need the vertexes to actually connect. It would be a general practice however to remove any faces you don’t want. So the face of the cylinder that is facing the cube could be removed and the cylinder simlpy moved so the lower edge goes below the face. That would be a simple thing to do.
There are other methods as well described above by makr and then to take it to an extreme for a more realistic molded or organic look you can visit my tutorial (link in signature) on susurf (subdivision surfaces/subpatch) modeling.
And to further explain. Also in some cases you don’t even want the parts to be one whole. Maybe in the real world the object is two pieces. It is better to model in two pieces if this is the case. So take a look at the real world object and see how it was made for a general guide as to how to construct it in computer 3D space.
This suggestion may not be that helpful in your current situation if you have already spent some time creating your cylindrical object first, but it may be helpful the next time you start a similar project from scratch. With this technique instead of joining the cylinder to the plane, you actually extrude the cylinder out of the plane. This will give a very smooth flowing join between the two shapes.
Start with your plane in edit mode then keystrokes"W", Subdivide Multi (4)
Select the middle 3 x 3 faces and delete.
Go to vertices mode and select the internal border vertices.
Press keystrokes Ctrl+Shift+S followed by 1 then enter. This is the “To Sphere function”.
Press “S” to scale the selection to size.
Now press “E” and select “Only Edges”. This will extrude the inner edge.
Now just smooth and subdivide to get the finish that you want.
I suggest the cur a hole in object tutorial here:
Your first method is very nice.
However I don’t understand your second method.
What do you mean by inset exactly. What does the mesh look like after the inset?
Are, for instance, the top left corners of the two planes connect via an edge? What is being subdivided?
@Richard Culver
That’s a very interesting tutorial on subpatch modeling.
This is also a very nice method.
That’s a very helpful video
>>What do you mean by inset exactly. What does the mesh look like after the inset?
<<
http://wiki.blender.org/index.php/Extensions:Py/Scripts/Manual/Mesh/Inset
Basically, you could say, it offsets (setbacks) each edge an equal distance, and cleans up the result. Works on faces in Blender.
To get a proper circle, using the “to sphere” tool, you’d need at least 16-24 vertices around the edges. so I’d suggested this: inset, subdivide the plane, inset, subdivide the innermost face - so you have the required no. of verts on the edges on the innermost face. Move the edges so that they form a square; then Ctrl+Shift+S (to sphere) to make it a circle.
Of course, Esemkay’s method should work equally well, besides being simpler 
Besides. there’s all the other methods. Good luck.
thanks makr,
I could not follow your explanation very precisely. But at least my result is probably similar to what you meant.