Integer: any whole number that has no fractional or nonzero decimal component. The integers are the numbers 0,1,2,3,... and -1,-2,-3,...
Natural number: all the positive integers (i.e: 1,2,3,...)
Prime number: A natural number which cannot be written as the product of two smaller natural numbers. For instance, 10 is not prime since 10=2×5 but 7 is prime since we can only have 7=7×1 (7 is not greater than 7)
Coprime: 2 numbers are coprime if there is no natural number (not equal to 1 or either number) that divides both of them. For instance, 9 and 14 are coprime. However, 10 and 14 are not coprime since 2 divides both 10 and 14.
Prime factorization: A way to express any natural number as a product of prime numbers. For instance, the prime factorization of 30 is 2*3*5 since 2, 3, and 5 are prime. Another example is the prime factorization of 231, which is 3*7*11
Rational number: a number which can be written as the ratio of 2 integers. For instance, 0.5=1⁄2 and -0.89=-89⁄100 are rational
Irrational number: a number which is not rational. For instance, √2 and π(pi) are irrational (here, √ means "square root". i.e, √x is a number which, when multiplied by itself, gives x)