A conjunction is a word that is used to combine two elements into a more complex phrase or sentence.
わたしは じでんしゃと へるめっとを かった。
I bought a bike and a helmet.
This groups "a bike" and "a helmet" to work as the direct object for "bought".
Here is another example:
やまださんと たなかさんは パーティーに いきました。
Mr. Yamada and Mr. Tanaka went to the party.
This groups "Mr. Yamada" and "Mr. Tanaka" to work as the subject for "went".
So far, we've joined nouns together to make a compound subject or a compound object.
Let's take this to the next step.
ひろくて しずかな いえに ひっこし する つもり です。
I plan to move to a roomy, quiet house.
Here, the equation isn't as obvious. In Japanese, you use the "conjunctive" conjugation of the adjective to combine it with another. In English, you place a comma after each adjective except the one immediately preceding the modified noun.
Here's another use of the "conjunctive" conjugation, only this is the conjugation for a verb.
ともだちの むらかみさんは パーティーに いって さけを のんだ。
My friend, Mr. Murakami, went to a party and drank sake.
Here we have a compound predicate formed from "went to a party" and "drank sake". Japanese doesn't use a conjunction per se because we used the conjunctive form of the verb; English doesn't have a conjunctive form for its verbs, so we use the conjunction "and".
In English, "but" is used as a contrastive conjunction. Similar words include "however" and "although". Japanese uses 「けれども」, 「けれど」, 「けど」, 「でも」 and simply 「が」.
わたしは パーティイーに いきたかったが くるまは うごけない。
I wanted to go to the party, but my car wouldn't start (lit. "the car couldn't move").
The English conjunctions "and" and "but" are called "postpositives" because they should never occur at the beginning of a sentence. Similarly, you should not find 「けれでも」, 「けれど」, 「けど」, or 「が」 as conjunctions at the beginning of a sentence.
For the most part, this covers the English conjunctions and some examples of how they relate to Japanese. Another related topic would be relative clauses, which are clauses that relate a piece of information about the topic they relate to (much as this clause relates to "relative clauses").