Because this is an innate spell given, is there a number of times that I can use it before I have to take a short or long rest? Using only once seems a bit odd.
Once per long rest as racial trait.
When you are level 5 or above you can use the darkness spell once per long rest as racial trait.
If your class allows you to cast darkness, you can do so with your spell slots.
The racial trait:
[...] When you reach 5th level, you can also cast the darkness spell once per day. [...]
There was an errata which says the following
Drow Magic (p. 24).
Here “once per day” means you must finish a long rest to cast the spell again with the trait.
Once per Long Rest
The Player's Handbook Errata clarifies this, reading:
Drow Magic (p. 24). Here “once per day” means you must finish a long rest to cast the spell again with the trait
This is a point that has come up across all three books, where 'once per day' has almost universally been clarified to mean 'once per long rest'.
Somewhat frustratingly, errata, and other rules clarifications are lumped in with all other announcements in the features section of the D&D part of the WotC website.
It does says you get to use it only once per day, and that's it:
Drow Magic. … When you reach 5th level, you can also cast the darkness spell once per day.
The errata for the PHB clarifies further that “once per day” means only once between long rests.