You learned to count when you wer little. When you became three you could hold up three fingers to indicate your age. The numbers you used then 1,2,3, ... we call counting numbers.
We add zero which represents no cows. Then we add one less than zero which represents "not only do I have no cows but I owe my brother a cow." We call that negative one. The counting numbers and zero, and the negatives of each counting number we call integers.
Now imagine a small gathering at an Italian restaurant. Nine guests show up. Three pizzas are ordered. The waiter cuts each pizza into six pieces. Each guest eats 2 pieces. We say that each guest ate one third of a pizza. We call numbers that are represented as one number divided by another rational numbers because they are based on ratios.
There are numbers which exist that are not rational numbers. We can find rational numbers that are very close to these numbers but can prove that the number is not a rational number. The number which when muliplied times itself equals two is such a number. We call it the square root of two, and it is not a rational number. Here is the proof. We call numbers like these irrational numbers because they cannot be represented exactly by any ratio. We call the integers and the rational numbers and the irrational numbers real numbers.
The subject of square roots brings up an ineresting subject. What is the square root of a negative number? we know that two times two is four, but now that we think about it, negative two times negative two is also four. So when someone asks the square root of four, the proper answer is "your choice of two or negative two."
Which leaves us with the question "What is the square root of a negative number?" Some genius figured that the square root of any negative number was just the square root of negative one times the square root of the coresponding positive number. Another genius decided to call the square root of negative one i. We really know that the square root of negative one is our choice of i or negative i. We call numbers containing i imaginary numbers because although they are useful they are not numbers that we can model with a length.
Now that we know that imaginary numbers exist, we can imagine adding real number to an imaginary number. We call those numbers complex numbers. Like imaginary numbers, they can be quite useful in some cases.
There are other types of numbers that have been defined but they are not needed for algebra, so you can ask Mr. Google.