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This word is annoyingly and confusingly used with different meanings in English language teaching.

1. Language functions (also called pragmatic functions) are the things we do with language, such as promising, inviting, giving advice, asking for permission, etc. Exponents are the actual pieces of language used to express that function.

2. When looking at a particular grammatical form which can be used in different ways with different meanings, these different meanings or uses are also called functions. It is very common for tenses to have multiple functions, for example:

a) He will get up early when I am trying to have a lie in.

b) He will get up early tomorrow because he has to catch a train at 7.30.

c) I expect he will get up early tomorrow because he's so excited.

In these examples, the future simple: He will get up serves the functions of expressing a) annoying habit/insistence b) future as fact c) prediction.

It is function in this second sense that is referred to when we talk about teaching the function of new language in a grammar or functional language lesson.

3. Grammatical function can refer to either the grammatical category a word in a sentence belongs to (See Unit 4, Part 1 Parts of speech):

Smoking is bad for you. - Smoking is a noun
They were smoking.  Smoking is a verb.

We had a party on the beach. On is a preposition.

or the role a word or phrase serves in the structure of a clause (see Unit 4, Part 2 Clause structure):

Smoking is bad for you. - Smoking is functioning as a subject.

We had a partyon the beach. - On the beach is a prepositional phrase functioning as an adverbial.

/'ʒɒnrə/ /ˈʒɒ̃rə/ The type and style of discourse (spoken or written), e.g., a letter of complaint, a romantic novel, a business email, newspaper article, promotional material, etc.

A gerund is the -ing form of a verb when it acts as a noun as in 'I love dancing,' and 'Studying grammar is enormously good fun.'

These activities are used at the start of a course to help the students and teacher get to know about each other. They are also called icebreakers.

It is the general meaning of a text.

This sound /ʔ/ is common in spoken English though it is not included in the phonetic chart used in English language teaching. Most obviously it is used when /t/ sounds are 'dropped' in words, such as 'bottle' or 'what,' but also occurs in other locations, e.g., something /'sʌʔmθɪŋ/. 
It is actually a kind of stop (consonant) produced by stopping the airflow with the glottis.
The sound is often regarded as low status and incorrect though it is in fact widely used by most native speakers.

See language grading.

Grammar refers to the rules which govern the way the words in a language change and how they are combined with other words in sentences.

It is a traditional approach to language teaching which focuses on learning grammar rules and applying them in order to translate texts from one language to another.

Also referred to a parts of speech. 

These are the names for the different classes to which words are assigned depending on their function in a sentence.

The parts of speech in English are the verb, noun, adjective, adverb, determiner, pronoun, preposition, conjunction, and interjection.