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.
Here is a brief history to help your understanding and I will get to your question at the end.
Binary is a little weird because we are so used to using a base 10 number system.  This is because humans have 10 fingers, when they ran out they had to use a stick, toe or something else to represent 10 fingers.  This it not true for all cultures though, some of the hunter gatherer populations (such as the Australian Aboriginal) used a base 5 number system (one hand) as producing large numbers were not necessary.
Anyway, the reason base 2 is important in computing is because a circuit can have two states, low voltage and high voltage; think of this like a switch (on and off).  Place 8 of these switches together and you have 1 byte (8 bits).  The best way to think of a bit is 1=on and 0=off which is exactly how it is represented in binary.  You might then have something like this 10011100 where 1's are high volts and 0 are low volts.  In early computers, physical switches were used which the the operator could turn on and off to create a program.
Nowadays, you will rarely need to use binary number in modern programming.  The only exceptions I can think of is bitwise arithmetic which are very fast and efficient ways of solving certain problems or maybe some form of computer hacking.  All I can suggest is learn the basics of it but don't worry about actually using it in everyday programming.
How do "binary" numbers relate to my everyday programming?