There are several different kinds of that in English (besides the ordinary demonstrative this, that, these, those). All of them are used in linking clauses, but they link different kinds of clauses, and follow different syntactic rules.
These that's are not, however, conjunctions; they are Complementizers. Complementizers are one of the parts of speech that have been discovered since the list of POS was drawn up in the 5th century. Not surprising, really; why should we use Roman science in the 21st century?
There are three different kinds of subordinate clauses, depending on what the clause is acting like:
- Adverb clauses normally contain a subordinate conjunction.
When I arrived, he was already here. He was already here when I arrived.
That used to be a possible complementizer for all tensed clauses,
as in Chaucer's Whan that Aprille ... hath perced ... 'When April has pierced'
or Andy Griffith's before that we set up the tent in What it Was, Was Football
That is used as a complementizer in tensed adverb clauses in Modern English only in local dialects like Griffith's North Carolina; that's what makes Chaucer sound strange. It is, however, used in tensed complements and relative clauses.
- Noun clauses (also called Complements or Verb Complements) act like nouns, as subject or object for a predicate. The clauses are boldfaced below. There are four varieties, each with its own complementizer:
Subject: For me to leave early would be a mistake. (infinitive complementizer)
Subject: Leaving early is not recommended. (gerund complementizer)
Subject: What he told me is not for publication. (embedded question complementizer)
Subject: That I have to leave early is unfortunate. (that complementizer)
Object: They told me to leave early. (infinitive complementizer)
Object: I hate his playing the piano at all hours. (gerund complementizer)
Object: I didn't hear what she told you. (embedded question complementizer)
Object: They told me that I had to leave early. (that complementizer)
Note that one of the complementizers is that, for tensed clauses.
This that may be deleted, provided it is not the first word in a sentence.
- Adjective clauses modify nouns or noun phrases. There are two kinds: relative clauses and NP complements; that occurs in restrictive relative clauses, and also in NP complements.
Relative: the man that came to dinner (that is subject; cannot be deleted)
Relative: the man that I saw in the station (that is not subject; may be deleted)
NP Complement: the rumor that I have to leave early (NP complements are just complements that modify nouns formed from verbs or predicate adjectives.)
It's possible that she needs a job ~ the possibility that she needs a job
That in NP complements can be deleted, but not nearly as often as in verb complements.