The brown spots you see on kitchen utensils -- particularly stainless steel knives -- are due to rust. The spots have two main causes; one is that something has been sitting against the metal of the utensil and creating rust, and the other is that the surface of the utensil has begun to corrode, and the pit is now rusting. Stainless steel seems very solid, but it is able to corrode and stain. Unfortunately, the corrosion isn’t always detectable until it starts to rust.
Stainless steel knives have a particular problem in that the qualities that make the stainless steel more resistant to rust also make it more difficult for the knife to cut well. The Cutlery and Allied Trades Research Association notes that the stainless steel used for knives is less resistant to corrosion than that used in spoons and forks. This is why you tend to see those spots on knives and not really on other utensils. If you leave a knife soaking in water, you increase the chances that the surface might develop pits that can rust later.
Thoroughly washing and drying the utensil, and ensuring it doesn’t rest against other items for long are the only ways to prevent non-corrosion rust spots from forming. Even leaving the utensil sitting in a pan or in the sink can create the conditions necessary for rust formation. Avoid soaking the utensil to prevent the surface from corroding. Remember that corroded pits are often very small. Just inspecting the surface once in a while and deeming it corrosion-free, in hopes of easing up on the discipline required to take care of it, isn’t going to work.
If the brown stains are relatively new and you’ve just noticed them, try mixing a little baking soda with water to scrub the rust off. Place the utensil on a paper towel, sprinkle the utensil with baking soda, dampen another paper towel and scrub. Be very careful if you are cleaning a sharp knife; you don’t want to slip and slice your fingers. Wash the utensil to remove any leftover particles and dry it thoroughly with a cloth or paper towel. If the rust is old, or is in a microscopic pit, you might not be able to get all of the spot off. Another option is a commercial rust cleaner meant for stainless steel. Whatever you use, do not use steel wool to scrub off the stain. The Cutlery and Allied Trades Research Association warns bits of steel wool can actually add brown spots to utensils. Heather Solos of Home Ec101 recommends using a kitchen scouring pad instead.