"Elf" is an English word for a family of supernatural beings - often called social supernatural beings or supernatural beings of nature - that are widespread throughout Europe. Those that hang most closely together can be found from Ireland to Sweden and Brittany to Britain and Iceland.

Because of the diverse geographic area, including a variety of languages and cultures, traditions about these supernatural beings exhibit a wide variety of characteristics. Even within a single culture, they can be described as having a variety of features. For example, your reference to the leprechaun brings to mind a specific form of Irish cobbler fairy (sidhe in Irish). Similarly, the banshee (woman sidhe; a grieving harbinger of death) is another type of Irish fairy. In addition, the Irish describe their more generic elf/fairy/sidhe in a wide variety of ways (human-sized or small; beautiful or less so, etc.)

The diversity that easily comes to mind in Ireland is magnified across the region. Names for these creatures include piskies/pixies, elves, fairies, sidhe, and huldre, and even the word troll (which speakers of English may regard as referring to an ogre-like creature) can be used for these supernatural entities, depending on the region in Scandinavia.

These entities can be seen as human-sized or diminutive. They are generally believed to be capable of invisibility and/or of transforming into other shapes. They were extremely dangerous and because they were capricious and easily offended, human contact was not advised and usually resulted in harm. That said, sometimes respectful people of good character could emerge from contact with something beneficial.

The greatest dread focused on these entities was their penchant for abducting people. This was particularly feared because abduction generally meant an eternity of servitude with the human deprived of heavenly salvation.

A variety of authors and cultural expressions have dipped into the great "stew" of beliefs and characteristics of elfin-like entities for a variety of purposes. Santa's elves combine numerous threads, but the Scandinavian nisse or tomte (with a variety of other names) seems to have contributed most to our understanding of Santa's helpers. These were diminutive household elfin spirits who were generally more helpful than their forest-dwelling cousins. The nisse/tomte was often associated with the winter solstice, and so it was easy for them to slide over to become affiliated with Santa - who is, in general terms, a larger, winter solstice elf who has a positive relationship with people in their house.

It appears that Tolkien borrowed heavily from the Welsh Tylwyth Teg, elves who were more court-like in the popular imagination, featuring kings and queens. The Tylwyth Teg (the fair family) tended to be regarded a human-sized - or nearly so. And they were exceedingly beautiful - hence the name.

Describing the differences of elfin-like creatures throughout Northern Europe has occupied many volumes over the years. Because of the rich diversity of traditions, it has been possible for many authors, commercial enterprises (Keebler elves and underpants gnomes!), and modern traditions to assume different, seemingly contradictory forms. Just as the folk traditions were at once built on a shared foundation and yet were different in specific details, so too are the modern manifestations.

Let me know if you have further questions.

edit: thanks for the silver!