"that" in Mandarin joining clauses or sentences?
The word &quot;that&quot; in your third example actually doesn't introduce a relative clause. Rather it's being used as a subordinating conjunction. In this situation, the word &quot;that&quot; is generally optional in English, and it also is similarly omitted in Chinese:

They said [that] they would come.
他们说他们会来。
Tāmen shuō tāmen huì lái.

Relative clauses generally describe a noun or noun phrase (known as the antecedent). In Chinese, while 的 de is often used to indicate possession, it more generally is used to link a description to a noun or noun phrase, and thus can similarly be used to introduce a relative clause. When the noun or noun phrase being described is understood in context, it is omitted, leaving simply description + 的.
Let's look at your first example:

He is the one that I saw.

The relative clause is &quot;I saw&quot;, which is simply 我看见 wǒ kànjiàn. In English the noun being described cannot be omitted when &quot;that&quot; is used to introduce the clause, which is why the word &quot;one&quot; appears to serve as the placeholder. In Chinese, it can be omitted, so &quot;the one that I saw&quot; ends up being simply 我看见的 wǒ kànjiàn de, thus:

He is the one that I saw.
他是我看见的。
Tā shì wǒ kànjiàn de.

If instead there was a specific noun, such as &quot;person&quot; instead of &quot;one&quot;, you would have:

He is the person that I saw.
他是我看见的人。
Tā shì wǒ kànjiàn de rén.

Applying the same principles to your second example:

It's the one that I want.
它是我想要的。
Tā shì wǒ xiǎngyào de.

EDIT - Addendum:
Note that in English the noun being described can be omitted if you instead use a relative pronoun to introduce the relative clause rather than using the word &quot;that&quot;. So instead of &quot;He is the one that I saw&quot; and &quot;It's the one that I want&quot;, you could say &quot;He is who I saw&quot; and &quot;It's what I want&quot;. The former have what are known as bound relative clauses while the latter have free relative clauses. The word &quot;that&quot; can only be used in bound relative clauses.
I just wanted to add this note to let you know that when you see a free relative clause, you can translate it to Chinese similarly, e.g.:

I like what he sees. (i.e., I like the one that he sees.)
我喜欢他看见的。
Wǒ xǐhuān tā kànjiàn de.