Lordosis is also called sway back. It can be the result of a birth defect or can develop later in life. The obvious symptom is a deep dip in the back - but surprisingly, horses with lordosis often perform normally (although fitting a saddle to them can be challenging). In other species, this level of spinal deformity would result in disability.
The difference is that a swaybacked horse still had a spine that follows a more or less smooth path, meaning the spinal cord isn't affected.
Congenital lordosis is genetic. Lordosis in older horses is most often caused by a long, weak back that eventually dips under the stress of work or, in the case of mares, carrying foals.