The subscript here indicates the base, also known as the radix. Typically, we deal with radix ten, or decimal, tied to an Arabic numeral system -- which is why we have ten digits. This developed because humans have ten fingers (or, as they're called in anatomy, digits ;-), so it was natural to base around.\[0, 1, 2, 3, 4, 5, 6, 7, 8, 9\]Past civilizations have used other choices for their radix and numeral system; in ancient Babylon, the sexagecimal system, which uses radix sixty, was used, with numerals for each. This is why we often speak of multiples of three and six, in things like sixty seconds per minute, sixty minute per hour, sixty arcseconds per arcminute, sixty arcminutes per degree, three-hundred-sixty degrees per circle... the list goes on and on.
Anyways, the key here is that different bases use different numerals. For convenience, we reuse our Arabic numeral system for bases that are less than ten. This is why binary, which is often talked about when speaking of electronics and computers, which uses radix two, employs 0, 1 as its numerals -- they are the possible values of an individual bit, or binary digit. Since binary is not easy to deal with, often people use hexadecimal, using radix sixteen, to represent values. They make use of the first six letters of the alphabet to supplement the ten we use, coming up with 0 to 9, followed by A to F. Below are examples of how to interpret numbers in different bases.\[173_{10}=1\times10^2+7\times10^1+3\times10^0=100+70+3=173\\FE_{16}=15\times16^1+14\times16^0=15=240+14=254\\1011_2=1\times2^3+0\times2^2+1\times2^1+1\times2^0=8+2+1=11\]Note that the term base might remind you of exponentiation and logarithms. That is, it's what is being taken the power of -- just like what is being done here.
Now, the actual relevant stuff is easy. Adding in foreign bases is just like you'd normally add, only you have to keep in mind that there is a different number of numerals, so explicit care must be taken to make sure you properly carry.\[44_5+11_5\]The first thing we do is add the last digits of both.\[\ \ \ 1+4=5\\\text{however we are in base 5, so...}\\\ \ \ 1_5+4_5=10_5\\\text{which means we need to carry.}\]Now, lets try to add the next digit, making sure to remember the digit carried over.\[10_5+10_5+40_5=110_5\]Again, we have a digit to carry over, so our result is ultimately...\[100_5+10_5+0_5=110_5\]Now, to check, let's rewrite them all in decimal and add quickly.\[44_5=4\times5^1+4\times5^0=24\\11_5=1\times5^1+1\times5^0=6\\44_5+11_5=24+6=30\]\[110_5=1\times5^2+1\times5^1+0\times5^0=25+5+0=30\]So, it all checks out :-)