It wasn't that long ago someone noticed the abbreviation for "March 10" was often expressed as MAR 10. Someone then realized that's how an elite 1990s hacker type would spell "Mario." Thus Mario Day became a thing, sort of like how May the Fourth became a big day for Star Wars. There's nothing actually linking either Mario or Star Wars to those dates specifically, but there's a neat little trick of the English language that makes it work.
In terms of Nintendo Switch deals, Mario Day is an opportunity to pick up some first-party Nintendo games at great prices. Mario Kart Home Circuit is also on sale, both the Mario and Luigi versions, and various retailers have also marked down prices on Mario-themed accessories and apparel.
TL;DR - Best Nintendo Switch Deals
Switch OLED In Stock at Amazon
Best MAR10 Nintendo Switch Deals
Nintendo Switch Game Deals at Amazon
Nintendo Switch Game Deals at Walmart
Nintendo Switch Game Deals at GameStop
Nintendo Switch Game Deals at Best Buy