Converting bases might be more intuitive.
For any number in any base, the value is = to the base to the power of the place * the number in the place done n times for every place in the number.
Take the number 101 in base 2:
( 1 * 20 ) + ( 0 * 21 ) + ( 1 * 22 ) = 5
Take the number 100 in base 10 (the base people most commonly use):
( 0 * 100 ) + ( 0 * 101 ) + ( 1 * 102 ) = 100
This works for the reasons that /u/timshoaf explained, however knowing how to convert bases is extremely helpful.
Additionally, this works the other way. Let's say that we wanted to convert the number 10 in base 10 to base 2. We take away powers of two until we've encoded the number. Ex:
10 - 21 - 23 = 0, THEREFORE 10 in base 2 is 1010.
Bases are important in computer science for a few reasons.
Computers use these things called bitwise operators to do a lot of arithmetic (typically denoted as >> or <<). Let's say we have the binary numbers 101 and 1010. Adding them is as simple as "turning switches on and off".
0101 = 1 + 4 = 5
1010 = 2 + 8 = 10
1111 = 1 + 2 + 4 + 8 = 15
This is incredibly useful when things have exclusively on and off states. (Think wires).
You may have heard of something called "hexadecimal" before, and it's exactly what it sounds like. "Hex" being 6 and "decimal" being 10. Base 16. Why base 16? Memory space. Think about it: 162 - 1 is the same as 28 - 1. Most computers are either 32 bit machines or 64 bit machines, which function similarly.
You can read more about base 16, base 2, and base 12 (the one everyone should use, the Babylonians used it, our clocks use it) from online sources. Feel free to message me if you want more information. Happy learning.