Elementals in Berserk

The existence of elementals in Berserk is revealed in volume 24, while Guts' group stays at Flora's place. Four elements are initially presented, corresponding to the four classical elements of the European tradition (dating back to ancient Greece). Their names and symbols are derived from the writings of Paracelsus and are associated with Alchemy.

In Japanese, the word "elemental" uses the kanji 元素霊 with エレメンタル added as furigana.

THE FOUR CLASSICAL ELEMENTS

Sylph (シルフェ) is given as the name for wind elementals (風の元素霊) and Salamander (サラマンデル) as the name for fire elementals when Serpico and Isidro receive their new equipment in episode 203 (Elementals). In volume 25, episode 212, as Schierke enters a trance to summon a spirit, we learn the word Undine (ウィンディヌ), for water elementals (水の元素霊).

One important thing to point out is that Miura uses somewhat nonstandard spellings for these names. For those with trouble reading Katakana, ツルフェ gives SHIRUFE, サラマンデル gives SARAMANDERU, and ウィンディヌ gives WINDINU. 

ツルフェ corresponds to the French version of the word, which means that its correct transliteration should be "Sylphe". I'm not quite sure about サラマンデル, but I'm guessing the spelling is still "Salamander" and it's just the pronunciation that differs (i.e. not "salamandah" as you'd say in English).

ウィンディヌ is a bit of a special case as it does not exactly correspond to any language I can think of. Using French, which it seems close to, it would transliterate to "Ouindine" instead of the standard "Ondine" (オンディーヌ). Honestly, Miura may have simply wanted to make his own version of the word.

We know earth elementals (地の精霊) exist as early as volume 24, but we don't actually get confirmation of their name until episode 343, at the beginning of volume 39. It is, without surprise, Gnome (ノーム). Unlike the previous three terms, this one follows the English pronunciation of the word, as opposed to, say, the French one (グノーム). Given that 15 years had elapsed since volume 24, Miura may have not been using the same references anymore by that point.

These constitute the four common elements of the Berserk world. Their symbols, shown above, are the same as in European alchemy. They can be seen when Schierke casts the formation of the four cardinal points, but also on Flora's golems or on magical equipment. Fun fact: they are part of the Unicode standard, so they can be rendered as text characters. Here they are: 🜁 🜃 🜂 🜄

BARYTES

That's it for the four classical elements, but there's more. In episode 346, Miura introduced one last type of elemental: バリュテース (BARYUTE–SU), the elementals of "weight" ("重さ"の元素霊). We write it "Barytes".

They are described thus: "They're the tiniest ones, and they connect the four others. They are so small that [Gedflynn] can't even see them with [his] spiritual sight. However, when they gather, they make things heavy, they darken the sky and bring on depression or obsession to the mind." They are also revealed to have a bit of an affinity with dwarves (sullen by nature), but not with the inhabitants of the forest of Elfhelm. And they are the key to flying on a broom for witches.

The name "Barytes" itself is interesting. It comes from the Greek word "βαρύτης", itself derived from "βαρύς", which means "heavy". It was used to name the mineral called "barite" in the real world, but that has no relation to the element in Berserk, and the pronunciation of the word is different.

As far as the properties of Barytes in Berserk go, they don't really fit those of "æther", which has had a bunch of notions attributed to it as a hypothetical fifth element over the centuries. It's pretty clear Miura made his own thing specifically for his world. There is no doubt more to Barytes that had yet to be revealed in the story.