In the number system we normally adopt in modern-day western life,
1 + 1 = 2
But the 1, the +, the = and the 2 are all just symbols, like letters in the alphabet. We use the same alphabet as that used in French, but letters put together that mean something in particular in English usually don't mean the same thing in French - often may just be 'gibberish' in fact, unless one recognises that it is a word from another language.
So we need to 'translate' this statement so that the equality makes sense. I'm going to pick the number symbols as the salient point, though there will be other ways of interpreting the collection of symbols.
So when would putting two 1s together give 11? Now we would automatically read '11' as meaning 11 in our numberworld, but you can look at it just as two 1 symbols next to one another. If I write it now as 'I' and 'II', this should ring a bell in your mind.
In Roman numerals (used back in the days of the Roman empire, and still used today, particularly on clocks), '1' is usually written as 'I' (a capital I), but these look rather similar don't they? (single upright symbol, with some embellishments, which we can easily recognise could mean 'one of', as it is a very plain symbol - debatable, given). So, in Roman numerals '1' is written as 'I' and '2' is written as 'II', so that I+I = II.
It is rather rare to see arithmetic in Roman numerals, as after the first few numbers it gets rather different to the Arabic number system we typically use now. The Roman systems uses several bases rather than our one base (10) - I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, M for 1000, and as soon as a number is closer to a base higher than it than one lower than it, it begins to be written as an amount less than that higher base, rather than an amount more than the lower base.
For example, 49 is written as IL, ie 1 less than 50. Also, dates can sometimes be given in Roman numerals. Now that we are in the 2nd decade of the 21st century, dates are being written as
MMXI, MMXII, MMXIII (2011,2012,2013)
As 2014 is 1 less than 2015, 2014 is written as MMXIV, and 2015 as MMXV.
In the lower case, Roman numerals are used for writing lists/bullet points (you'll see an option in your word processor) - i), ii), iii), iv), v) etc
This way of referring to the numbers from above as well as below makes arithmetic difficult (in particular, adding up in columns - try it!), so in modern times we have abandoned using the Roman number system for arithmetic in favour of the arithmetic. However, if Rome had conquered the world as it came fairly close to doing, we might still be using something that looks more like the Roman number system than the one that we do use.
There are other possible interpretations, but I would say that 1+1=11
because we are doing one plus one in the Roman number system, that is I + I = II (replacing the symbol 1 with the symbol I, which represents the same thing as well as looking very similar in shape)