It is a German word, it means “hill culture,” and in practice is a method of gardening where you bury wood and then plant on top of it.
Things like “lasagna gardening” (where you build up a raised bed by alternating layers of compost, twigs & debris, newspaper and cardboard, and soil) are a variation on this, but true hugelkultur is pretty extreme, not just laying in sticks and twigs and garden debris with soil as you build a raise bed, hugelkultur could best be describes as covering up a firewood pile with soil, so you have a mound, or a swale, or a terrace. You put whole logs down, piling them on top of each other in a long triangular shape, then cover with soil. You have this sort of mound garden. Why do this at all? The wood both increases drainage and water retention, buried wood is capable of holding water like a sponge, it becomes a buried reservoir of water for your plants, and as it decomposes it increases organic matter and microbial activity in the soil and creates air pockets. It is also a decent method in any area where a raised bed is a good idea, namely areas where the ground is too rocky, the soil too shallow, or the soil just of too poor quality to do a proper garden bed. So rather than dig a bed into the ground, you build a bed on top of ground.
Hugelkultur beds don’t have to be huge piles, they can also be built flat. You could layer logs and other wood at the bottom of a flat bed, like a raft of wood, then pile soil on top. That works too. As the wood decomposes it will of course shrink and so you need to account for that.
The flat hugelkultur bed is the method I followed. I used the branches from a tree that was removed out of the large garden bed space. I also used the forest behind the house to forage for downed branches.