Conjunctions are part of several named grammatical constructions: "compounds", relative clauses, et cetera. It's important to be familiar with all of them, because conjunctions are essential to making complex sentences.

"Compound subjects" are subjects made up of more than one noun:

The King and Queen laughed at me.

"Compound predicates" have more than one verb:

The aliens pointed and laughed at me.

"Compound adjectives" are made up of more than one adjective:

My mean and stupid sister laughed at me.

"Compound adverbs" are made up of more than one adverb:

Victoria cruelly and mercilessly laughed at me.

I think the pattern is clear by now, but there are also compound pronouns like him or her and prepositions like inside and outside.

Conjunctions are really important!

Coordinating conjunctions combine two or more words, phrases or independent clauses. The items [words, phrases, or independent clauses] are of equal importance in the sentence. (This is the key difference between coordinating and subordinating conjunctions: subordinate conjunctions are followed by a clause that is less important to the meaning of the sentence).

There are only four regular coordinating conjunctions in English: and, or, nor, but.

And

Remember that "and" joins two or more things of the same type. You can write "noun and noun" or "verb and verb" but not "verb and noun":

Tom likes to run and to swim.

Tom likes to swim in lakes and rivers.

❌ Tom likes to swim and rivers.

Also remember that when you combine more than two things, a comma goes between all items except the last two:

Tom likes to swim in lakes, rivers, oceans and volcanoes.

Some people, however, prefer to also use a comma between the last two items (before the conjunction). This is also fine.

Tom likes to swim in lakes, rivers, oceans, and volcanoes.

Or

The rules for using "or" are the same as the rules for "and".

Sasha or Masha will probably win the race.

I like to eat cookies before breakfast or after lunch.

Nor

Used when combining two or more things that are both negative or excluded. Often (but not always) combined with the conjunction, "neither":

Neither Bob nor Angelika won the race.

If nor is not used with neither, there must be some other "negative" word used in the sentence to partner with nor:

Bob did not win the race, nor did Angelika.

Sometimes nor can be seen on its own. In these cases it refers back to something "negative" in a previouse sentence:

"I don't like pancakes," said Bob.
"Nor do I," said Angelika.

But

"But" is used to show a contrast between two things. A key difference between "but" and "and" and "or" is that "but" can only join two items.

I like dogs but hate cats.

❌ I like dogs, birds but hate cats.

"But" can also be used as a subordinating conjunction!

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There are also several part-time conjunctions. These words are more often used as adverbs or prepositions, but are also frequently used as conjunctions. The following are just a few examples, not an exclusive list:

For

"For" is a little strange because it's usually used as a preposition, but sometimes it can be used as a coordinating conjunction. When it's used as a conjunction it has to come between the two things it's "coordinating", and it only works with clauses. "For" as a conjunction shows "causation" —the clause following the conjunction gives a reason or explanation for the first one.

I don't eat mushrooms, for I am allergic to them.

Yet

"Yet" is like "but" in that it shows a contrast between [only] two things, but its use emphasizes that the speaker is somewhat surprised:

My sister is young yet wise.

[We usually don't expect wisdom from young people.]

The rebels were losing yet kept fighting.

So

"So" can be used as an adverb (common) or as an adjective or pronoun (infrequent), but sometimes it's a conjunction. It generally shows that something is the result of something else:

I gave the parrot a cookie so he would shut up.

or shows that something is the reason for something else:

It's dark outside so we should go to sleep.

Subordinating conjunctions combine the main clause and a subordinate clause in a sentence. That might sound complicated, but it isn't really. It just means you can make sentences like:

I ate an apple because I'm a winner.

Note that the main clause is "I ate an apple", and the subordinate clause is "because I'm a winner". It should be clear that the main clause is more important. The subordinate clause only exists to provide extra details about the main clause (in this case, answering why?).

There are a few facts you should remember in order to understand subordinating conjunctions:

1. A clause is a group of words that has a noun and a verb, but isn't its own sentence. (Clauses are the parts that complex sentences are made of.)
2. Some sentences have two clauses: a main clause, and a subordinate clause. The subordinate clause is used to add details to (provide extra information about) the main clause.
3. Subordinate clauses start with a subordinating conjunction, like "but", "so", "because", "although", et cetera.
4. The subordinate clause can come before or after the main clause, but if it comes first it has to be followed by a comma.

Which just means that you can write

I gave my daughter a cookie because she was hungry.

or

Because she was hungry, I gave my daughter a cookie.

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Some subordinating conjunctions are used to show a cause-and-effect relationship between two clauses:

The poor man left a tip, though he didn't want to.

Conjunctions of this type include as, unless, since, though and because.

Other subordinating conjunctions show a relationship in time or space between two clauses:

I stopped buying fast food once I learned how to cook.

Whenever I see the moon, I cry.

Conjunctions of this type include once, while, wherever, whenever, before, and after.

Correlative conjunctons are pairs of words that combine to establish a relationship between two items in a sentence. Some common pairs are neither-nor, either-or, both-and and not only-but also. In the examples below, notice that the items joined by the conjunctions are of the same type / structure.

Neither you nor I know where Batman lives.

The dragon will either eat us or let us go.

Both vampires and werewolves hunt their victims at night.

I not only play guitar but also write songs about crocodiles.