If you look in the kitchen of the average family home, you will find an assortment of kitchen knives; unsurprisingly, the majority of these will be made of stainless steel.
Stainless steel has become the standard for domestic knives and they do a good job around a kitchen. If you just want to get dinner on the table, not think about your knives beyond whether they're just barely sharp enough to blunder their way through an onion, and not worry if you mistreat them to no end, then stick with stainless.
When it comes to the finer points of metallurgy, what's important to know is that the difference between carbon steel and stainless steel is that stainless has chromium added to carbon steel's basic iron-and-carbon mix. The chromium bestows upon stainless its resistance to corrosion and rust.
That rust resistance of stainless, though, comes at a price. First, stainless tends to be a softer form of steel, which means it often won't hold an edge as well as carbon steel. That's right, carbon steel stays sharper longer than stainless.
Carbon steel, despite being harder than stainless steel, is way easier to sharpen than stainless. This is critical because knives that are used frequently simply don't stay very sharp for long. Sure, they stay sharp enough to do their job reasonably well, and certainly sharp enough to cut you if you slip, but not high-performance, precision sharp.
If you want a professional knife that can be sharpened to a razored edge, carbon steel is your friend.
So here's the downside: carbon steel is more fragile. It's a more brittle material, which means it's more likely to chip if you drop it or toss it into the dishwasher. It also rusts and stains easily. With prolonged exposure, it can also react with certain foods like onion, darkening them. Because of its tendency to stain, in time it becomes more and more yours. Like good leather shoes or raw denim, carbon steel only gets better with time, developing a patina that tells the story of how it's been used. That's a truly beautiful thing.
Put simply; as well as developing a unique character, a good carbon steel blade which is looked after correctly will out perform any other knife in the kitchen.