My guess is
matter (or maybe "elements")
I Last billions of years or a couple of milliseconds.
There is matter (elements) that is stable and lasts billions of years, whereas some of the radioactive elements only last a few milliseconds before they decay
I embody the Jurrasic period and the Renaissance.
Matter literally embodies everything that is associated with these periods. They would not have "existed" without matter.
I am composed of 24 letters of the English alphabet, and 116 chemical elements.
The periodic table contains (approximately) 116 elements. (This changes occasionally. Right now, it's 118). The short forms of these elements (e.g. "H" for Hydrogen or "Mo" for Molybdenum) contain 24 letters of the english alphabet. Particularly, they contain all letters except for J and Q (I just checked this, just in case ;-))
I am your entire life, and I am waiting on you.
That's a bit philosophical, but a common definition of "life" relies on "matter". Not sure whether the "waiting on you" part should interpreted somehow.
The black image could be a pun,
related to "dark matter" or "black matter"....
(I'm not a native english speaker, so all this may be utterly wrong)