Solar system data API? I'm prototyping a real time solar system / milky way discovery app.

Could you please help me on:
-What is the best orbital data public API I can use?
-What is the standard numerical way to define orbit?

I'm looking for something like JPL Small-Body Database Browser, but for every celestial objcets, and / or missions.
Here is a short answer. @barrycarter's helpful comment links to a much larger list of resources. Also, you may find answers below [this question] helpful. 

There are Spice kernels for many (but not all) exploratory spacecraft as well, you can view them through the Horizons interface or find them via ftp. 

For most objects in Earth orbit, Two Line Elements (TLEs) from Celestrak or Space-Trakck are useful. See this answer as well.