character v. glyph v. grapheme/morphogram v. ideogram v. logogram v. pictogram v. symbol vs. syllable
In Chinese, the overwhelming majority of characters is monosyllabic, which is why the first two named columns are essentially identical. This is in stark contrast to Japanese for instance, where the following example shows how one character can bear as many as five syllables:
 
 (Japanese)        承        る
 (Hiragana)        うけたまわ　る
 (Transliteration) uketamawa.ru
 (Translation)     to attend to




In English, notice how the word renewables can be split into re-, new-, -able, and -s. They each mean something (some are of grammatical worth, such as the plurality marker -s), and cannot be split, according to meaning, any further. These chunks are therefore known as morphemes.
There is only one way to split bisyllabic (or bi-character; they are frankly interchangeable in Chinese) words — you split them in half. But not all bisyllabic words in Chinese are splittable.
When you split 鐵路, you see 'iron' and 'road' which makes sense.
When you split 蝴蝶, 蝶 means 'butterfly' (likely an abbreviation of 蝴蝶), but the meaning of 蝴 is illusive (perhaps nonexistent). The same goes for 珊瑚 ('coral'), 玫瑰 ('rose'), 駱駝 ('camel')... These words are called binomes (連綿詞). We write them with two characters they must be bound together to signify the desired meaning. Individually, they do not make any sense. An explanation for their unsplittability is that they perhaps were once monosyllabic in the past. Take 蝴蝶 as an example:

From Middle Chinese *ɦuo dep (first syllable unstressed), from Old Chinese *ɡaː l'eːb, derived from a proto-form of *kʰleːp ~ *ɦleːp, a prefixed form of the root *lep (“butterfly”). Wiktionary

Binomes are much rarer than splittable bisyllabic characters. In most cases Chinese words are splittable.

There are many askers on Chinese SE that are interested to know why a character bears a certain meaning, i.e., how its form (形) relates to its meaning (義). Here we put the questions under the tag glyph-origin. In Chinese, we use liushu (六書) for explanation (there are six possible categories):

Pictograms (象形): 山 ('mountain') really looks like the shape of a mountain.

Ideograms (指事): 一 ('one'), 二 ('two'), 三 ('three') do not look like any tangible objects. They are abstract realisations of concepts.

Compound ideographs (會意): 休 ('to rest') is a combination of the pictograms 人 (equivalent to 亻, meaning 'man') and 木 ('tree'). It represents a man resting by a tree.

Phono-semantic compounds (形聲): most Chinese characters belong to this group. If I were to create a new writing system for English using emojis and say

 'to believe'

I am using  only for the sound of the word (bee + leaf) and  for the meaning of the word (to pray).

Rebus (假借) and derivative cognates (轉注).



Grapheme is the smallest functional unit of a writing system. So c in cat and 朋 in 朋友 ('friend') are both graphemes. But a grapheme in an alphabetical language is not necessarily a morpheme, whereas in Chinese it's possible. Since language learners are usually more interested in the meaning of a word/character, rather than its form (how it relates to meaning is a separate matter), we seldom use this terminology.
A logographm or a logograph is a written character that represents a word or a morpheme. (I took this definition from Wikipedia, the Cambridge Dictionary of Linguistics does not provide a definition.) Again I think this is used in a meta-language sense: they include Chinese characters, Egyptian hieroglyphs, and cuneiforms. We seldom use these terms in Chinese SE.