Ethnically, the name is Scottish, but like every other surname the Stewarts have migrated literally everywhere. Indeed, as others have stated, the line originated in France with the Normans. The first Stewart being Robert FitzAlan. The name has a very humble beginning, originating from the two root words "Sty- Warden", a nice name for pig keeper. The name developed into one that reflected a "steward" or "exchequer". Since the Celts originally considered wealth by the amount livestock one owned a counter of livestock or a "sty warden" developed into a "steward" or in gaelic "Stuibhard", a fancy name for basically a tax collector.
On the Stewart coat-of-arms the exchequer past in evident in the 3 bands of white and blue checks which run across the shield.
The name would change in spelling depending upon where the Stewart migrated to. If they moved south and had contact with the Norman English the name would be spelled "Stuart" and so on.
If you know where in Scotland (or England) your Stewart family came from you can usually tell which sept or clan you can claim heritage from. There are many branches; Stewarts of Bute, Black, McAlpine, Atholl, Appin, and lots of people who believe they are Royals.....very small chance of that as a direct line but perhaps as a cousin.
Also, at least in my case, the spelling of my name has change back and forth 4 times in 300 years. So, before you decide that you know for certain check all the details!
Hope that helps!