The term 'grammar' can refer to two different things: descriptive grammar and prescriptive grammar. Generally, grammar is the set of rules that governs the structure and composition of the elements, such as words, phrases and sentences, in a natural language.
Descriptive grammar, in particular describes the rules that govern any form of a natural language, including its non-prestige dialects.
Prescriptive grammar presents a set of codified rules that govern the standard or prestige form of a natural language. It can also encompass orthographical rules regarding spelling and punctuation and is the basis of modern standard written language.
Proper grammar is essentially adhering to prescriptive grammar, particularly in written language, but also in spoken language. Standard English itself has several variants including Standard American English, Standard British English and Standard Australian English, though their prescriptive grammars are mostly similar, outside of pronunciation.
The subject of a sentence and the corresponding verb should agree in number. A singular subject should take a singular verb and a plural subject should take a plural verb. Subjects composed of two or more nouns adjoined with an 'and' take a plural verb. Two or more singular nouns connected by an 'or' or a 'nor' take a singular verb.
In subject composed of two nouns where one is a pronoun, the pronoun takes the nominal form. Use 'a' before words that start with any consonant, except silent 'h'. Use 'an' before words that start with a vowel, except for a glide 'u' or 'o'. For a word that starts with a silent 'h', use 'an'. Hard 'h' is preceded by 'a'. For words that start with 'u' or 'o' pronounced like 'y(u)' and 'w(o)', use 'a'. Style guides are excellent sources for learning Standard English grammar.