The words "even" and "odd" have several meanings in mathematics. If we're talking about numbers, an even number is a number that is divisible by 2 with nothing left over. Yes, it's a major bummer when there are no leftovers, but even numbers aren't really that delicious in the first place, so it's not a huge loss.
...-6, -4, -2, 0, 2, 4, 6... are even. 0 is even, because 0 can be divided by anything with nothing left over. For example, 0/2 = 0.
We can write any even number as 2n where n is an integer. The table below shows some even numbers:
If we were to continue filling in this table in both directions forever, we would get all the even numbers. Although, since there's an infinite amount of them, we'd need an awfully long scroll bar over there →
An odd number is any number that is divisible by 2 with 1 left over. (Great - that'll be tomorrow's lunch.) Any odd number can be written as 2n + 1 where n is an integer.
Every natural number must be either even or odd. If we pick any number and divide it by 2, we will either have 0 left over or 1 left over. There are no other possibilities. And that's not just us being pessimistic. It's a fact, yo.