The first coil that I designed used 1000 turns of wire. My wire had a resistance of 2.95 Ohms, and I think my battery has an internal resistance of about .05 Ohms, so the total conveniently came out to 3 Ohms so my 1.5V battery got me half an Amp, which multiplied out by the number of turns gave me 500 Amp-turns.
I decided that it might work better to use more turns of wire, so I increased to 2100 turns, and this time I got a resistance of 4.45 Ohms. Along with the battery resistance this means that I got only one third of an Amp, but multiplied out by 2100 turns this means that I got 700 Amp-turns. Since this coil has more Amp-turns, and it is about the same size as my old coil, I got a stronger electro-magnet.
I want the battery that is powering the coil to last for as long as possible. I calculated the drain on the battery by figuring out how many Watts of power I was drawing. The first coil used 0.708 Watts, and the second coil used 0.5 Watts. So even though the second coil is a stronger magnet, it uses less power from the battery! If you can figure out this sort of thing then you can build impressive electromagnets that can lift heavy objects without drawing too much power from a battery.
There are more formulas that you can use to figure out things like how much force you will get from your magnet. There are also software packages that can be used to predict how well your electromagnets will work and how much force you can get out of them. These programs can also predict how much torque a motor will have.