Grammar

Grammar is the study of words, how they are used in sentences, and how they change in different situations. The Ancient Greeks used to call it grammatike tekhne, the craft of letters. It can have any of these meanings:
The study of a language: how it works, and everything about it. This is background research on language.
The study of sentence structure. Rules and examples show how the language should be used. This is a correct usage grammar, as in a textbook or manual/guide.
The system which people learn as they grow up. This is the native-speaker's grammar.
When we speak, we use the native person's grammar, or as near as we can. When we write, we try to write with correct grammar. So, speaking and writing a language each have their own style.
Different languages
All languages have their own grammar. Most European languages are rather similar.
English makes few changes to its word endings ('suffixes'). In the Italic or 'Romance' languages (such as French, Italian, and Spanish), word endings carry a lot of meaning. In English we have just a few: plurals and possessives (John's) are the most common. In our verbs we have dropped most endings except one: I love, you love, but she loves. That final 's' comes from the Anglo-Saxon, which had more suffixes. Verbs do have endings which show changes in tense: walked, walking.
Word order is the other big difference. Romance languages normally put adjectives after the nouns to which they refer. For example, in English, a person may say I like fast cars, but in Spanish, it is Me gustan los coches rapidos. The order of the words has changed: if just the words, without the grammar, are translated into English, it would mean 'to me they please the cars fast'. This is because Spanish and English have different rules about word order. In German, verbs often come near the end of sentences (as: Die Katze hat die Nahrung gegessen), whereas in English we usually put them between subject and object, as: the cat has eaten the food.
Changing language
Written grammar changes slowly but spoken grammar is more fluid. Sentences which English speakers find normal today, might have seemed strange 100 years ago. And they might not, because many of our favourite sayings come from the Authorized King James Version of the Bible, and from Shakespeare.
Different people speak with grammar that differs from that of other people. For example, people who use the dialects called General American English and BBC English might say, I didn't do anything, while someone who speaks what is called African American Vernacular English or AAVE might say, I didn't do nothing. London working class version: I ain't done nuffink! These are called double negatives, and are found almost entirely in spoken English, and seldom written.
These differences are called dialects. The dialect a person uses is usually decided by where they live. Even though the dialects of English use different words or word order, they still have grammar rules. However, when writing in American English, grammar uses the rules of General American English. When people talk about using 'proper English', they usually mean using the grammar of general British English, as described in standard reference works. The models for spoken English in Britain are often called Received Pronunciation or BBC English.
Parts of speech
Grammar studies nouns, pronouns, verbs, adjectives, adverbs, prepositions, conjunctions, sentences, phrases, clauses, interjections.
Nouns
Nouns are 'thing' words like 'table and 'chair'. They are objects, things you see in everyday life. Proper nouns are names of specific places, people, or other things like days of the week. The name 'James' is a proper noun, as is 'Wednesday' and 'London'. Nouns can also be abstract things, such as 'suffering' or 'happiness'.
Verbs
Verbs are words that describe actions: "Ryan threw the ball". State: "I am worried". The basic verb form is called the infinitive. The infinitive for existence is "to be". A famous example is the speech of Hamlet: To be or not to be, that is the question.
Variations of the infinitive create verb tenses.
Past tense = was
Present tense = is
Future tense = will/shall
Adjectives
Adjectives describe nouns. For example, the pretty in "pretty bicycle" says that the bicycle is pretty. In other words, the "pretty" is describing the bicycle. This can also happen with a place. For example, the tall in "that's a tall building" is describing the building.
Syntax
Grammar studies syntax which is how the "parts of speech" fit together according to rules and create sentences. Sentences fit together and create paragraphs.