A neti pot (sometimes also called nasal cup or jala neti (जलनेति)) is a small gravy bowl-like pot designed to rinse the nasal cavity. Basically, you pour water (usually a saline solution) down one nostril, and it comes out the other together with the mucus (that is snot for you regular folks) and whatever else happens to be in your nose (such as a tape recorder). The practice has some real benefits — in medical literature it's usually referred to as nasal irrigation or saline nasal irrigation — but there's also a rather large amount of woo attached to it.
 Real benefits
There is decent evidence that it helps reduce the symptoms of allergies, sinusitis, rhinitis, and other nasal unpleasantries (such as the common cold), but it is important to realize that mucus does play an important role in our body's immune system, and long-term use is harmful and will increase the risk of rhinosinusitis (infected sinuses). Excessive short-term usage may also be harmful.
Short-term usage of a neti pot is safe, but only if the water has been boiled or adequately filtered (<1µm) before using—at least 2 deaths from the Naegleria fowleri amoeba have been reported.
The exact mechanism of efficacy isn't well understood, but it's probably due to the removal of mucus and/or irritants such as dust and pollen, meaning it's one of the few "detox" methods that actually works. Of course, it may not be possible to use when your nose is badly stuffed up, so even for its medically sound applications it cannot be a cure-all.
 Not real benefits
There is no evidence that the neti pot can also prevent diseases (such as the common cold), as is often claimed. In fact, long-term usage is associated with increased infections of rhinosinusitis (infected sinuses)
Using any liquid other than (possibly saline) water is also unlikely to provide additional benefits; some people apparently think it's a good idea to use cranberry juice or even urine. There's also a thing called "Super Neti Juice", which contains "elemental silver, de-ionized water, peppermint oil and a surfactant". The peppermint oil may make for a nicer experience much like peppermint in toothpaste does, but you can pick up a bottle of that for a few dollars; no need to shell out $45.
The full list of other health benefits touted by some woo-meisters is long, and includes (but is not limited to) weight loss, better circulation, smoke cessation, ear disorders, depression (and other mental illnesses), better smell, increased intelligence, better eyesight, fewer wrinkles, less grey hair, "cooling and soothing effect on the brain", "higher states of meditation". All of this has no evidence whatsoever and is simply made up.