What do the terms "syntagmatic" and "paradigmatic" mean?

Both syntagmatic and paradigmatic analyses contribute to the inherent meaning and understanding of a given language.
Syntagmatic relationships are created with the understanding that components of language fall in a certain order. Letters, words, and sentences in a particular order create meaning. This meaning is created from the lowest levels, stringing certain letters together and building to create meaning by sequencing paragraphs in a particular order. Sequence is a focal point in this area of thought.
Paradigmatic relationships look instead at the way groups of words relate to each other. Adjectives, therefore, can be substituted for each other:

We found a furry cat.
vs
We found a fluffy cat.

Pronouns are examined together:

I want pizza.
We want pizza.

All parts of speech are thus able to fall into similar subgroups, allowing for certain types of substitutions within these subgroups.
Being able to fluently interact with a language—including retrieval of information, creating meaning, summarizing text or events, and creating language with meaning—necessitates the use of both syntagmatic and paradigmatic relations.

These two terms refer to contrasting ways to analyze language in structural linguistics. Within a given language, every item has both kinds of relationship with other items.

The syntagmatic relationship refers to syntax or sentence structure. These structures put limits on the placement of items and, in doing so, help the listener or reader make sense even of sentences that contain unfamiliar words. English sentences use subject-verb sequence, such as "he is," so a word following a subject will often be a verb, as in "Dogs eat." Some verbs will also be followed by an object, such as "kicks the ball."


Paradigmatic relationships refer to a word's relation with the same type of word, one that can have the same function and be substituted for it in the same structure: replacing "dogs" with "horses" yields "horses eat."


Paradigmatic relationships also extend to qualities of words unrelated to their function or syntactical location. "Bin," "thin," and "win" are related by minimal sound (phoneme), but "bin" is a noun, "thin" is an adjective, and "win" is (usually) a verb.
https://www.encyclopedia.com/social-sciences-and-law/sociology-and-social-reform/sociology-general-terms-and-concepts/paradigmatic-and-syntagmatic