Mathematicians have many categories according to which they classify numbers. From the simple odd and even partition to more complicated like happy or prime or even recurring decimals and complex numbers. Let us put some order in this chaos and define some necessary categories of numbers.
Natural Numbers
Natural numbers are the most simple numbers, how mathematics started in a sense. The set of the natural numbers start with one then two, later 0 was added and then comes twenty nine trillion, eight hundred forty four billion, five hundred twenty million, four hundred twenty two thousand and six hundred sixty nine ( which is apparently too big for BigInteger) and so on and so forth. Natural numbers are measurable, because each unit is discrete yet they are infinite because they do not end, there is no last natural number and for every one number that claims it is the last you can always get the next by just adding one to it and it is still a natural number, guaranteed. Anyhow, then came the integer numbers, which are all the naturals and their opposites and then fractions and it only gets more complicated afterwards, so unless you are studying to be a mathematician or accountant (in which case you would probably not bother reading my humble site) do not bother too much.
Prime Numbers
Among the natural numbers there is a category of numbers which have a certain peculiarity that sets them apart from the rest of the numbers and allows them to stand out with their position in the line up of natural numbers. Those that pertain to this category are called prime and their peculiarity is that, apart from 1 and themselves, they cannot be divided by another natural number. One ancient Greek method to filter out the non primes is the Eratosthenes' Sieve and that is a good place to start your exploration of the prime numbers.
Decimal Numbers
Decimals started off together with modern civilization in big cities, like Mesopotamia, Egypt and Athens, and express fractions of variable units that needed to be expressed devoid of their wholesomeness. If one gets a pie and cuts it in 6 pieces and them eats 3, then he ate 1/2 or 0.500000 of the pie, whilst the rest goes for maintenance duties. Should one want to eat 6 pieces and give his dog a piece (probably from another pie) then he would have spent 7/6 of a pie or 1.17 or so. Floats, as these numbers are called in computing, require of the user to set up how much precision is required, which is dependent on the programming language, and may vary from 2 to 8 decimal digits, which O.K. for most practical purposes, excluding multi-threading on modern computers. Large or Double (precision) numbers are required from then on.
Binary Numbers
The systematic study of binary numbers probably started as a part of probabilistic computations which had to simply define the value of an outcome in a closed manner that would describe its state across many possible computations. Then again it might had been Socratic philosophers that after having exhausted infinite possibilities, like Zeno's paradox, had to certify truth value of assertions scientifically. On/off, 1/0, true/false are all ways to express boolean values, yet the mathematical form of Arabic numerals smaller than 2 is the principal way to express them.
It is not clear when it started as a mathematical discipline, yet binary arithmetics express the mathematical world in a prosaic yet elegant and machine understandable fashion: Every number can be represented by ones and zeroes and then by appointing the right addendum among the powers of two. 2 is 10, 17 is 10001, and -417.82 is -110100001.1101000111101011100001.