I am trying very hard to develop a much deeper understanding of programming as a whole. I understand the textbook definition of "binary", but what I don't understand is exactly how it applies to my day to day programming? 
The concept of "binary numbers" vs .. well... "regular" numbers, is completely lost on me despite my best attempts to research and understand the concept. 
I am someone who originally taught myself to program by building stupid little adventure games in early DOS Basic and C, and now currently does most (er, all) of my work in PHP, JavaScript, Rails, and other "web" languages. I find that so much of this logic is abstracted out in these higher level languages that I ultimately feel I am missing many of the tools I need to continue progressing and writing better code. 
If anyone could point me in the direction of a good, solid practical learning resource, or explain it here, it would be massively appreciated.
I'm not so much looking for the 'definition' (I've read the wikipedia page a few times now), but more some direction on how I can incorporate this new-found knowledge of exactly what binary numbers are into my day to day programming, if at all. I'm primarily writing in PHP these days, so references to that language specifically would be very helpful. 
Edit: As pointed out.. binary is a representation of a number, not a different system altogether.. So to revise my question, what are the benefits (if any) of using binary representation of numbers rather than just... numbers.
As a web guy, you no doubt understand the importance of unicode. Unicode is represented in hexidecimal format when viewing character sets not supported by your system. Hexidecimal also appears in RGB values, and memory addresses. Hexideciaml is, among many things, a shorthand for writing out long binary characters.
Finally, binary numbers work as the basis of truthiness: 1 is true, while 0 is always false.
Go check out a book on digital fundementals, and try your hand at boolean logic. You'll never look at if a and not b or c the same way again!
How do "binary" numbers relate to my everyday programming?