High Elves are the elves who once set foot in Valinor (one of the Undying Lands), where the Valar, including Elbereth, live, and then returned to Middle-earth. The history and categorisation of the different elves is very long and complicated, going all the way back to their first awakening by Ilúvatar before the Sun and Moon were created. Below is a very rough summary, but links are provided underneath for a more detailed account.
The elves were awakened in Cuiviénen in the far east of Middle-earth by Ilúvatar. Walking through the forests they found each other, and began to create language, music and poetry. Soon, the elves were discovered by the Valar and called by them to Valinor; some followed the summons whereas others remained behind.
At Valinor, one of the elves called Fёanor created the Silmarils (jewels), using some of the light of the Two Trees that shone on Valinor. Melkor (an evil Vala) stole the Silmarils and poisoned the Trees, for which he was named Morgoth (Black Enemy). Fёanor and the Noldorin Elves swore to return the Silmarils, and travelled back to Middle-earth to fight Morgoth. Eventually, with the help of the Valar, Morgoth was defeated. The Valar called the elves back to Valinor but some chose to stay, establishing realms in Middle-earth. However, all elves feel the call of the distant land, and will eventually cross the Sundering Seas to Valinor when they have tired of the mortal world. Many crossed after the war against Sauron at the end of the Second Age, and now the numbers of elves left in Middle-earth are dwindling.
For a more detailed account of Fёanor, the Noldorin Elves, and the Silmarils, see bookmark for page 253.