It's not that I doubt you -- no, not at all, because I have a copy of the original, so it's clearly a vector -- but, damn, is that really Illustrator? O.O This is why I will never be a real artist. No patience to figure out how the hell to make that software do my bidding. >.< Also, it's a small thing, but you win my respect for leaving his signature on it. Excellent job, a perfect tribute.
Patience is a virtue - especially when you're dealing with Illustrator. People want me to tell them how to do these, but there isn't much I can say other than to just open the program and fiddle with it until they've figured it out.
Vectors are digital shapes created in programs like Adobe Illustrator. You draw them via a system of points and lines that can be combined together to create shapes and whole images like this one. That's a very basic explanation of what they are.
The artist is sadly dead now. This piece was something he drew in the 1930s or '40s (the drawing says 1924, but it's generally thought that he added that date later). I worked from a print I have of the drawing to recreate it in vectors.
You can find his work all over the place, just Google Alberto Vargas.