A Landgrave (Landgraf in German, comes followed by all sorts of descriptors in Latin) was the holder of a hereditary feudal title that has no precise contemporary equivalent. The word is probably best rendered as count in modern texts, based on the French nobility ranks, but may be considered closer to an earl in the Anglo-Norman system. Landgraviates were created in the late 11th century, starting with the counties of Brabant and Guelders in Lorraine, and were peculiar to the territories of the Holy Roman Empire.
There existed a very important distinction between an ordinary count (Graf) and a landgrave. A count with the title of landgrave owed allegiance directly to the Holy Roman Emperor, not to a holder of an intermediate title. Landgraves had in their possession the sort of land holdings more commonly granted to dukes and earls. They also enjoyed a high degree of sovereignty. Indeed, in later years German/Holy Roman emperors found it politically expedient to elevate some landgraves to princes of the realm and made them de facto equals of dukes and such.
While the title was first used in the west of the empire, it became most significant further east as the empire's geographical balance shifted in that direction, to what is now central and southern Germany. Some of the most important landgraves were the rulers of Thuringia (Thüringen) and Hesse (Hessen), both of who acquired more titles over time. They also share the unusual property of having one of their ancestors and 13th century rulers (St. Elizabeth) as their common patron saint. Less prominent landgraves ruled over smaller parts of Alsacia, Swabia, and Bavaria, particularly during the 13th and 14th centuries.
The title of landgrave, along with all other German titles of nobility, has been legally extinct since the beginning of the Weimar Republic. The heirs of the house of Hesse are the only ones who still style themselves as Landgraves. The title is used only by the head of the house, and that's only because it's the last title left after the ranks of elector and archduke disappeared following the abolition of the Electorate and the dissolution of the Empire. The rest of the heirs modestly call themselves princes. Indeed, the proper title of the spawn of any landgrave was prince, as it was for many ranks of the German nobility. Which reminds us that most of the princes in our German-derived fairy tales are an artifact of translation and very few of them are destined to be kings. Bummer.