It is common to use water as an analogy to electricity because the analogy works so well. We'll use it here, too.
Electricity in a battery is like water in a water tower. To get higher water flow through your pipes you must make the water level in the tower higher. The increased pressure forces more water through the same size pipe. To get increased electrical current to flow in a wire you need a battery with a higher voltage. It will force more electrons through the same size wire. Voltage is the pressure which forces current to flow.
Another way to get increased water flow is to make the pipe larger. A larger pipe has less resistance to water flow and so more water flows at the same pressure. If you use a larger wire size, more current will flow with the same battery voltage because larger size wire has lower resistance to electron flow.
You can vary the amount of water flow by varying the resistance to flow using a faucet. When the faucet is open part way, less than the maximum water will flow. Putting a resistor in the electrical circuit cuts the current flow to less than the amount that would normally flow through the wire.
If you have problems when you think about electricity, remember to consider water in a pipe. Sometimes it is easier to visualize what is happening when you compare it to something you are familiar with.
The field of electronics is based on the manipulation of current flow to achieve a desired result. In a toaster you make the current flow through the heating elements to get them hot enough to toast bread. In a television you manipulate the current flow to herd little dots into an image. In an Arduino circuit, you manipulate the current flow to do anything you can think of. And it is all just as simple as water flowing through a pipe.