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.
rather more of an experience rather than a solid answer:
actually, you don't actually need binary because it's pretty much abstracted in programming nowadays (depending on what you program). binary has more use in the systems design and networking.
some things my colleagues at school do in their majors:

processor instruction sets and operations (op codes)
networking and data transmission
hacking (especially memory "tampering". more of hex but still related)
memory allocation (in assembly, we use hex but sometimes binary)

you need to know how these "regular numbers" are represented and understood by the machine - hence all those "conversion lessons" like hex to binary, binary to octal etc. machines only read binary.
How do "binary" numbers relate to my everyday programming?