Ok so I am learning about induction and I always see coils of wire with no insulator. So I was wondering, What is the significance of it being a coil of wire rather than just a solid cylinder since there are already many points of contact between each "wrap" of the coil?
The part you are missing is that what looks like uninsulated wire actually isn't. A lot of enamel coated or "magnet wire" can look like bare copper at first glance, but the wire is actually coated with a thin semi-transparent insulation layer. The reason for using thin insulation is so that lots of turns of the coil can fit into the tightest possible space.
Electrically, a coil of wire is quite different from a cylinder. To make a magnetic field, you need current flowing around where you want the field. Think of a cylinder of current surrounding the area. The magnetic field is proportional to the total current in the cylinder.
A coil is sortof a cheat on that. The same current is re-used each turn to add to the apparent cylinder current. Let's say you have a coil with 100 turns. If you run 1 A thru it, each of those turns contributes 1 A to the overall cylinder current. You get the same magnetic field as if it were a solid cylinder with 100 A running around it.
Wire used in coils usually has an insulating coating - enamel or similar.
As long as that is intact, the coil does not approximate a solid conducting cylinder (which would be a single "turn" of very thick wire), but instead a multi-turn coil where the magnetic field sees the current multiplied by the number of turns and the inductance varies as the square of the number of turns.
The difference is that electrons follow the path of least resistance. For a solid cylindrical conductor that would be "straight through." For a coil of wire the current must flow "round and round" in space. The importance of this is related to what's going on physically here (ala Maxwell's Equations). Current flowing through a conductor induces a magnetic vector field that "wraps around" the direction the current is flowing in. The topology of a coil is such that these magnetic fields are superimposed so as to create a net magnetic field that has a kind of "flows out" of one end of the coil and "wraps around" back into the other end. Without going into a whole lot of math, it's this composite magnetic field that I believe gives an inductor the properties we are used to thinking about in circuits (i.e. resistance to changes in current). It's also this net magnetic field where the energy is effectively "stored" in an inductor.
Re-reading your question, it is essential that the windings of a coil not "touch" electrically or you will not get the effects I described above. They may in fact only "look" like they are not insulated, but in fact they have to be, as Chris pointed out.