Lumbar radiculopathy is irritation of a nerve root as it exits the spine. "Radic-" is just Latin for "nerve root" and "-opathy" is just "pathology." So "radiculopathy" is "pathology of a nerve root." Commonly, people refer to it as a pinched nerve in the back.
Lumbar radiculopathy can arise from a variety of factors within the spine. You can have a herniated disc, there can be some arthritis in the facet joints, you can have bone spurs, you can have spondylolisthesis where the bones in the spine actually slip on top of one another, you can have the ligamentum flavum buckling. All of these things can contribute to narrowing the space where the nerve exits the spine. Essentially, if you think of the hole where the nerve comes out, and as that hole gets encroached upon, there is less room for give within the hole, it doesn't have to be a mechanical constriction of the nerve, but there is less room within the hole for the nerve to exist and there is more of a propensity towards an inflammatory response. That inflammatory response can cause pain, numbness, weakness in the leg. Often the pain will radiate into the leg and that's why we call it lumbar radiculopathy.
Diagnosing lumbar radiculopathy, just like diagnose most things medically, begins with a good history and a good physical examination - this is really Medicine 101. Often from the history and physical examination, we can tell whether or not a lumbar radiculopathy is present and we'll typically get an MRI to look at the anatomy to see what anatomical processes are contributing to that radiculopathy - herniated disc, spinal stenosis, foraminal stenosis, facet joint arthropathy. All of these things can contribute to it, so we look at the anatomy to get a sense of which level and what the underlying pathophysiologic process is that is actually going on.
Sometimes if the diagnosis is in doubt or if we want to get a sense of the extent of the radiculopathy, we may also get a nerve test, which is a way of testing the nerves to see where any inflammation is coming from a little more exactly and to what extent the inflammation is there and to rule out the more peripheral causes that can sometimes mimic a lumbar radiculopathy. Nerve testing is commonly referred to as an EMG.