Roots and affixes are types of morphemes. Words can be divided into morphemes, and each morpheme has some meaning. Some morphemes indicate grammatical content, but English is not a heavily inflected language, so most morphemes in English have some lexical content, that is most morphemes have a "meaning".
Some morphemes are "free". A free morpheme can function as a complete word. As an example of a free morpheme consider the word "happy" It is a free morpheme. Morphemes that are not free are said to be "bound". A bound morpheme must be combined with other morphemes to make a word. In the word "unhappinesses" the morphemes "un-", "-ness" and "-s" are bound morphemes.
In heavily inflected languages there are few free morphemes, as most words require some affix. However there is often one morpheme that carries the main lexical meaning. This is the "root". The bound morphemes that attach to the root are "affixes". In the Latin "amo" (I love), "am-" is the root and "-o" is the affix. In English, most roots are free morphemes.
So in the word "unhappinesses", there is a root, "happy", a negative prefix "un-", a suffix "-ness" that forms a noun, and a grammatical suffix "-s" to indicate a plural noun. If you know the meaning of each of these four morphemes, you can work out the meaning of "unhappinesses".
Now attaching affixes is one way to create new words. "Happy" is an example. The etymology of "happy" is "hap" (an old word meaning "luck") and the affix "-y" that forms adjectives. But it would be wrong to say that "happy" is formed from a root word "hap". The word "hap" is no longer a morpheme in "happy". You cannot understand The division of "happy" into "hap" and "-y" is part of the etymology.
When a word is derived from Latin or Greek, it may have Latin or Greek affixes. Sometimes free morphemes in Latin can become affixes in English. But it is normally best to think of the word as being a single free morpheme. There can be a grey area. For example "Kleptocrat" is a recent coinage that was formed, in English, by combining a (somewhat mangled" Greek word meaning "thief" with the suffix -crat. On the other hand "aristocrat" is derived from a French word, and the meaning of "aristo" as an independent morpheme has been lost. "Aristocrat" is essentially a single free morpheme, not a root-affix word.
For learners of English, understanding roots and affixes can help rapidly expand your vocabulary. It is essential that learners know about and are able to use affixes like "un-" and "-ness" But you must beware of words like "aristocrat" that appear to have a suffix, but are actually single free morphemes. Learning the etymology of words is an interesting part of English, but it isn't essential. You can be a very good English speaker and have no knowledge of etymology. See Does understanding Greek as well as Latin help for improving English vocabulary? for more on this.