The reason that you want to build a coil is to make an electro-magnet. You want the magnet to be as strong as possible in order to operate your machine. The type of electro-magnet that you should build is a made from a large number of turns of wire wrapped around an iron post.
The amount of magnetic field you get from this type of magnet is proportional to the number of turns, so you want a lot of turns. The field is also proportional to the amount of current in the wire, so you want lots of current.
Until you reach a limit called saturation, the field from your magnet increases as the product of the number of turns and the current. This is often abbreviated as NI, (N times I) where N=the number of turns, and I=the current through the wire.
We want the electro-magnet to have a certain size so that it fits in your machine. So for a given size coil, how do you maximize NI?
If you use a very thin wire (like the size of a human hair) you can fit many turns in your magnet, so N is very large. If the wire is too small though, it will break when you wind the coil. The more turns on the coil, the more problem there is with breakage. Try wrapping a string around one of your fingers. As you add more turns, it feels tighter and tighter on your finger. This same stress is what will snap the wire in a coil if there are too many turns and the wire is too small.
The other problem with small wires is that finer wires have larger resistance, which means that a higher voltage is needed to get a given current. This is in accordance with a famous formula called "Ohm's Law." Ohm's law is simply V=IR, where V is the voltage, R is the resistance, and I is the current. So if you have a 1.5v battery in your circuit, V=1.5. If your wire has a resistance of 10 Ohms, then the current is I = V/R = 1.5/10 = 0.15 Amps. If you use thicker wire, you could get a resistance of 1 Ohm, so you would get 1.5 Amps. The thicker wire, however, would mean that you couldn't get as many turns.
Why not just use a really short wire to get a very small resistance and use a huge amount of current? How about 0.001 Ohms? That should get 1500 Amps? This actually doesn't work, because the battery has a small amount of resistance inside. The resistance of the battery might be 0.1 Ohms, so you can't get more than 15 amps from it. This internal resistance is smaller in car batteries, which can deliver huge amounts of current. Still, don't put a wire across your car battery to see how much current you will get, because it will start a fire. You can find how much internal resistance a battery has by reading the battery datasheet.
Speaking of fire, why not just use a large voltage across the coil to get more current? The problem is that the energy that will be released in the form of heat has the formula E=V^2/R. (That's energy (Watts) equals voltage squared divided by resistance.) So more voltage causes more heat. Too much heat will melt or burn something! You have to be careful that the battery does not blow up. Ordinary alkaline batteries will not normally explode, but NiCad batteries are notorious for catching fire if they are short-circuited. This is especially bad since NiCads contain Cadmium, which is toxic. So don't use NiCads unless you are sure that you know what you are doing. I know what I'm doing, and I still hate NiCads. They just aren't worth the trouble except in very special circumstances, such as radio-controlled electric-powered planes.
The things to keep in mind are: