Binary is just another way of counting. Almost every single number you encounter in everyday life is in base 10. In base 10, you count each digit up, from 0 to 9. When you add 1 to 9 though, the 9 in the ones place changes to a 0, and the 0 in the tens place changes to a 1. So when you have 099, and you increase it by one, the 9 in the ones place changes to a zero and adds 1 to the 9 in the tens place, which then changes to a 0 and adds 1 to the 0 in the hundreds place, giving you 100. If you have a number like 5473, that means you have 3 ones, 7 tens, 4 hundreds, and 5 thousands. The number of place the the left the number is is 10^n, where n is the number of places to the left it is. So the ones place is 10^0, the tens place is 10^1, the hundreds place is 10^2, the thousands place is 10^3, etc.
But in binary, you only have two digits, a 0 and a 1. So when you have a 1 and add another 1 to it, you need to shift up a place, to get 10. In binary, 1 + 1 = 10. With base 10, you have 10^n, but in base 2, you have 2^n. So the first place is 2^0, the second is 2^1, the third 2^2, the fourth 2^3, etc. So with a binary number 101101, you have one 2^0 (1), one 2^2 (4), one 2^3 (8), and one 2^5 (32), which gives you 45.
These are both different types of positional notations for numbers. There are other types of positional systems, one for every real number in fact, but some other common types include unary (base 1), quinary (base 5), octal (base 8), hexadecimal (base 16), and duodecimal (base 12). The first known positional notation was made by the Babylonians, and was base 60. Hexadecimal is actually used in computers as well, with each possible group of 4 binary digits signifying one hexadecimal digit. The 6 digits past 9 in hexadecimal are A through F. This makes it easier to write numbers when dealing with computers, since binary numbers use too many digits for relatively small numbers. An example of a non positional notation would be Roman numerals. With Roman numerals, you need new digits the higher you go up, which makes it hard to symbolize large numbers. With positional systems, you can literally represent any possible number.
The reason humans use base 10 is thought to be because we have 10 fingers. But some cultures (like the Babylonians mentioned before) use different bases. Some parts of Africa use binary, some Aboriginal Australian languages use a quinary system, the Native American Yuki and Pamean languages use octal systems, and some Nigerian and Indian languages use duodecimal.
The reason computers use binary is because using only two different voltages is much less prone to error and easier to construct than using any more than that. However, if we develop technology so it’s possible to make some type of higher base computer which has mechanics that can compute just as quickly and compactly as the system we use now, it will be able to process things with much less information (for example, if we develop a base 10 computer, the binary string 10000000 will become 128, saving 5 places). Boolean algebra is used in computers in the form of logic gates to manipulate binary numbers.