This is a great question, without the magnetic field Earth could not support life as we know it. We would have no ocean or air to breathe. More on that later but first on the how the core produces a magnetic field.

Geologist hypothesize that the core of Earth is composed of a solid iron core and liquid iron outer core. Now two things are important about the cores's composition that makes it magnetic: 1. it is composed of iron and 2. it has a liquid outer core. As you likely know iron is a magnetic element. From physics we know a magnetic field can be induced when a charged ion moves in space. Think of it like electricity the power lines have flowing electrons in them as they move from the power plant to your home they actually induce a magnetic field in the power line.

Now for Earth's liquid iron outer core, it is so hot the iron exists in a liquid, ionically charged state. So when the charged liquid iron moves about in the outer core the material induces Earth's magnetic field.

Earth's magnetic field allows all life to exist as we know it today. Without our magnetic field Earth would be much like Mars, the magnetic field extends into outer space beyond our atmosphere and deflects high energy particles emitted by the sun. If these high- energy particles were not deflected they would strip Earth's atmosphere, all the oceans would evaporate into space, it would get very cold below freezing, and destroy all life as we know it.

The Earth’s outer core is made of a mixture of elements. The three main constituents are iron 85% Nickel 10% and probably Sulfur 5%.

Because the Earth’s outer core is liquid, and because it is electrically conducting, the natural response of a moving conducting fluid is to set up a magnetic field; this is called Faraday’s law and was studied by JC Maxwell in the late 1800s and early 1900's.

As you acknowledge in your second question, iron makes the earth's core so magnetic. I think we know that it is iron that makes the earth's core so magnetic, because we know how big the earth is, and we know how fast the earth rotates, so we can probably calculate how heavy the earth is.

Then, we know how strong the magnetism is in different materials, so we can calculate how much of each material would be needed to make a planet that is as big and heavy and magnetic as earth.

Does that make sense?
Keep asking questions!

Good question!
Short answer: nobody knows.

Long answer: It has something to do with the fact that molten iron, which is much of what the core is made of, is capable of being magnetic, and because there are currents that flow through the molten core. Nobody really knows what the directions and strengths of these currents are, as well as other things that might matter, such as how much nickel and other stuff there is in the core as well. I am not enough of a geochemist or a physicist to tell you what the latest theories are, however, but I can tell you that this is an area of active research.