Spelling is always different, I go with the most common.
Hadji Jalili is one of the more notable designers/workshops (slightly unclear history here) for weaving Oriental Rugs in the late 19th century Tabriz Persia. Traditionally, rugs of which are attributed to his name have been of very fine quality. Other than quality, the eye for proper expert design mapping was one of the most distinguishable aspects of such rugs which set them apart.
There were many types of designs they wove, prayer rugs, traditional medallion, etc. Many of such examples will feature light colors, with beautiful aqua blue, ivory and very soft earth tones. Some of the larger rugs I have seen are often on a burnt orange background - Claremont rug company of CA has some pieces they consider to be Hadji Jalili if you want to take a look under their Tabriz carpets, although I think they use the term a little loosely myself. When I come across some I will post.
SpongoBongo.com has some great examples, but notice how they put "Hadji Jalili" in quotes for a lot of the Sotheby's descriptions. Most reputable sellers/museums, collectors will not say definitively that a rug is such unless there is some sort of inscription or they have had their rug in posession for a great deal of time and it has always been called as such. The problem is most of these rugs have no documentation. If the seller uses the term very loosely to a low quality rug such as the rug on eBay, the jist is similar to selling a replica or a fake in a sense. The thing is, you usually can say
what is not a Jalili rug - but stating
what is can be very difficult.