What makes a circuit digital?
First of all, I like the title of the question, "What makes a circuit Digital" ;).

The world around us is analog so the electronics started with exploring the analog components like the ones mentioned by you resistors, capacitors etc. The need for digital circuits was realized when we started building devices like computers, started storing information digitally and many other things.

If you look closely, digital world is based on voltages similar to the analog world. The only difference and the most advantageous feature of digital is that we have reduced the number of variable to just 0 and 1. 

Consider a voltage range 0V to 5V.Lets assume that 0V-2.5V is X and 2.5V to 5V is Y. Here we have grouped the voltages in two blocks but the bottom it remains the same. Sam we did with 0 and 1. To make it convenient for ourselves we thought (just for fun!) let X be 0 and Y be 1 and it all started from there. Whether you get a voltage of 1.1V or 2.4V we will interpret is as a 0. Similarly,you get 3.5V or 4.9V it is a 1 for us. This is biggest advantage of digital circuits called Noise Margin.    

Since your question was about how a circuit becomes digital, i don't want to get into the technicality much. but if really want to blame someone for this. get hold of this man Claude Shannon. It is believed that it was his article that led to birth of Digital Revolution.