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As you can see in the images, I have run the Select Non Manifold command to check for any errors. I got an error for the whole 3D text which is attached to the block of mesh behind it. I joined them together using Ctrl + J, but I don't know how to fix the errors.

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  • $\begingroup$ A blend file will be really helpful to solve it. $\endgroup$ – Leon Cheung Jun 8 '16 at 3:32
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I believe that's not an error, it's doing what you're exactly asking it to do.
Joining meshes using Ctrl + J do not affect their geometry, it just puts them as one object, think of it as a way of grouping meshes, any doubles or holes will remain the same, so what you actually did is overlapping your meshes, what you need to do is to close the holes, remove doubles, in other words make the mesh manifold.
Trying to understand better what a manifold mesh is, think about it as a model of an open bottle, with a single surface, the hole at the top makes it non manifold, since the "exposed" inside of the bottle, has no dedicated faces, faces normals are pointing outside, while adding a Solidify modifier, or closing the hole, makes it manifold (in case there are no doubles) because all "exposed" faces now have normals pointing outside the volume of the mesh itself.

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As explained in ermod's answer there are several types of non manifold meshes, and as such many types of errors that could be happening with those faces.

Understanding which case is present is key to solving the problem. When invoking the Select non- manifold operator you can choose from several options in the Tool Shelf so you can pinpoint which one of them selects those faces so you can determine how to solve the problem (like boundary faces, non contiguous, wire, etc.)

If those were created from a text object then most likely your problem is disconnected faces (a case of Boundaries type of non manifold).

Joining with Ctrl + J only joins the meshes into a single object, it does not alter the mesh or merge geometry together.

You can solve try to solve it all blindly by just doing a brute force Select All > W > Remove Doubles and then Ctrl + N > Recalculate Normals, althoug actually understanding what is wrong with your mesh would be a more "correct" and desirable course of action.

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From the manual:

Non-Manifold meshes essentially define geometry which cannot exist in the real world. This kind of geometry is not suitable for several types of operations, especially those where knowing the volume (inside/outside) of the object is important (refraction, fluids, booleans, or 3D printing, to name a few). A non-manifold mesh is a mesh in which the structure of a non-overlapped surface (based on it's connected faces) won't determine the inside or the outside of a volume based on its normals, defining a single surface for both sides, but ended with flipped normals. When working with non-closed volumes, a non-manifold mesh will always determine at least one discontinuity in the normal directions, either by an inversion of a connected loop, or by an odd number of surfaces. A non-manifold mesh will always define an odd number of surfaces.

There are several types of non-manifold geometry:

  • Some borders and holes (edges with only a single connected face), as faces have no thickness.
  • Edges and vertices not belonging to any face (wire).
  • Edges connected to 3 or more faces (interior faces).
  • Vertices belonging to faces that are not adjoining (e.g. 2 cones sharing the vertex at the apex).

http://3dprintingninja.blogspot.com/2014/12/non-manifolds-manual-methods-of-fixing.html

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