The magnetic effect is exactly the same in a copper wire and a coil wire. When a current passes through a wire, the current induces a magnetic field around the wire. The magnetic field has a certain direction associated with it. In the case of the current-carrying wire, the direction of the magnetic field happens to be tangentially to an imaginary circle drawn around the wire.

When the wire is twisted into a geometry of the coil, the magnetic fields induced by each "turn" of the coil either add up or subtract out because of the direction of the magnetic field from different turns are different in some regions and the same in some other regions. It turns out that the magnetic field due to a current-carrying coil of wire looks like the magnetic field due to a bar magnet with north and south poles.

The magnetic effect of an electric current depends on the shape of the wire, not the material that the wire is made out of. A coil of wire concentrates the magnetic field within the coil, producing a strong, local, magnetic field. A simple length of wire does not produce such a concentrated or strong field.

For both the straight and the coiled wire, you can use the "right hand rule" to determine the shape of the magnetic field resulting from current flow in the wire. Pretend that you are grasping the wire with your right hand and point your right thumb in the direction of the conventional current (positive charge flow, I). The direction that your fingers curl gives you the shape and direction of the magnetic field lines.

I included some pictures here . See if you can use the right hand rule to reproduce the shape and direction of the magnetic fields in the two cases. Notice how in the case of the coiled wire, the field directions "add" inside and outside of the wire. The result looks like the magnetic field lines that would be produced with a simple bar magnet.