Below is a list of formation methods and simplification methods mentioned in the Etymology section in the Advanced Pop-up.
The character originated as a picture of an actual object., such as 日 or 羊, which are pictures of the sun and a goat, respectively.
Associative Compound (会意字 / 會意字)
The character is composed of two or more components which together convey the desired meaning, such as 坐 (sit) which is composed of two people 人 sitting above the ground 土.
There are two types. The first is a character that is an arbitrary creation used to depict an abstract concept, such as 上 (above) and 下 (below) or 一, 二, 三. The second type uses a combination of a picture and indicator mark to point to the area being defined, such as 本(root)，which is a picture of a tree with an indicator marking the area where the root is.
Pictophonetic (形声字 / 形聲字)
The character is composed of at least two components and one of them provide an indication of the prnunciation while the rest provide an indication of the meaning, such as 按 (to push), where the 安 component indicates that the character is pronunced like "an" but its meaning has something to do with the hand 扌.
Phonetic Loan (假借字)
The character originally had a completely unrelated meaning but because there weren't enough characters at the dawn of writing, scribes borrowed other characters that had the same pronunciation. In many cases, the character's original meaning is no longer associated with the character or has been lost or the original meaning is now expressed by a modification of the original character.
For example, the character 因 (as in 因爲） currently meaning "cause" is a phonetic loan as the character originally indicated some form of mattress. Since 因 took on this new meaning, the original meaning is now expressed by 茵, where the 艹 grass radical is used to re-clarify that the character is about a grass mattress.
Mutually Explanatory (转注字 / 轉注字)
This category is rarely used. For example the characters 考 kǎo "to verify" and 老 lǎo "old", which had similar Old Chinese pronunciations and may once have been the same word, meaning "elderly person", but became lexicalized into two separate words. The term does not appear in the body of the dictionary, and is often omitted from modern systems.
Created as unique simplification
One-of-a-kind simplifications which cannot be generalized to other characters even if they contain the simplified character as a component.. Example: 個 » 个, 後 » 后. As the second example illustrates, the simplified character may not be an entirely new character but the adoption of an existing character with a similar sound and giving it a new meaning in addition to its previous one.
Promoted a historical variant
to be the simplified form. Example: 掛 » 挂, 兇 » 凶.
Created as generic character simplification
or derived from all characters containing the simplified character as a component (unless already covered by the unique simplification rule). Example: 貝 » 贝 and all characters containing them, such as 員, 費, 貴 becoming 员,费, 贵, etc.
Created as generic radical simplification
or derived from all characters containing the simplified character as a component (unless already covered by the unique simplification rule). Example: 言 » 讠and all characters containing them, such as 說, 話, 許 becoming 说, 话, 许, etc.