The A series of paper sizes are designed so that when you cut one in half, you get two sheets of the next-smallest-size, and every size has height and width in the same proportions.
An A0 piece of paper is exactly 1 square metre, which gives a width of 84.1cm and height of 118.9cm (to the nearest mm).
An A1 sheet of paper has a length that's the same as the width of A0 sheet of paper. So you can get two sheets of A1 paper by cutting an A0 sheet in half.
The sizes keep halving all the way down to A10:
A4 is the closest to the traditional size used for correspondence, making A4 the globally excepted standard paper size outside of the United State.
Having standard paper sizes like these is helpful when making a booklet of a particular size, from a sheet of the next-largest-size folded in half. So A3 sheets of paper can be folded to create a 4 page A4-size booklets. This could be folded in half again to make an 8 page A5 booklet.
A consequence of having standard paper sizes like this is that it becomes easy to calculate the weight of single sheets. Standard photocopy paper is usually 80gsm (grams per square metre), thus an A0 sheet, being 1 square metre, weighs 80g. An A1 weighs 40g, A2 is 20g, A3 is 10g, and A4 is 5g.
Here is a great video explaining paper sizes by Number Hub with Matt Parker.