Hi, Caitlin!
"O" scale is 1:48, or 0.250 inch to a foot. It is one of the "older" scales in model railroading, invented by the Ives Co. in 1910.
It was actually
smaller than some of the other scales that were out, and maybe because of this became very popular.
"HO" scale is 1:87, or 0.138 inch to a foot. It is close to (but not exactly) half the scale of "O", so it is "Half O", or "HO." This is the most popular scale today.

Yea us !!!!!!!!
My understanding is that HO was developed in the 1930s, when the Depression caused tight budgets for model railroaders.
As I can hear some of the N-scalers in the background, I will point out that "N" scale is one of the newer scales, at 1:160. At about half HO scale, it seems to be increasing in popularity with those who can work with such a small model scale. I have trouble enough with the small parts in HO scale, sometimes!
There is even a "Z" scale at 1:220, but you need a microscope to see it !!!!!!!!!!
(just kidding)
At the opposite end, there is "G" at 1:22.5 - these are the big trains you usually see at some Christmas displays.
Hoep this helps!
- George