A Gaylord box is typically used only a few times, depending on what you've used it for. One that was used to hold old clothing might remain usable for a while, but one used to hold heavy, discarded computer monitors might end up with too many gouges and tears after one use to be used again. Whatever the case, when that box becomes too damaged to use, you have to get rid of it. It's cardboard, so you won't have to trash it, but your options for boxes this big are somewhat different from what you might be able to do with a basic book box.
Truly Recycling the Cardboard
You do have the option of recycling the cardboard with a recycling center. Break the box down — be careful of staples — and take the item to a recycling center if you don't have a recycling dumpster large enough (or empty enough) to hold the box. It can be annoying to cut the box down into small parts that would fit in a smaller recycling container, and it's much easier to throw the cardboard into the back of a truck and head toward the local recycling place. However, not all of these places will let you redeem the cardboard for money. If you want to get paid for the cardboard, you'd need to look at selling it.
Selling the Cardboard, Sometimes Labeled as Recycling
When you look up places to recycle cardboard (if you don't already have a regular place to go), you'll see a lot of companies that talk about recycling when they really mean that they'll buy the boxes from you. Check with these companies first to see if they're just buying the cardboard (in which case, the damage isn't much of an issue), or if they mean they want to buy the intact box. If they want an intact box, ask about transport.
What to Do With the Pallet
Some Gaylord box companies also sell pallets, and these can be reusable for a long time. However, if you no longer want the one you have, you can recycle the pallet at any recycling company that takes wood products. Some companies buy used pallets, too, but breaking these down and recycling them as scrap wood may be the fastest option if you need to free up space on your property.
Gaylord boxes and other bulk boxes are big enough that you do have to decide what you want to do with them fairly quickly. Luckily, the options are easy to take advantage of.