An article is a word that is used with a noun. There are two kinds of articles: definite articles and indefinite articles.
In English there is just one definite article: the. There are two indefinite articles: a and an. The word an is used before a word starting with a vowel sound (which does not have to be a vowel letter): we say a horse, a child, a European (European has a "Y" sound, /j/, which is a consonant sound), a university, but an orange, an elephant.
Some languages have more than one word for the. In some languages, this is because each noun is either masculine or feminine or, in some languages it can be masculine, feminine or neuter. For example: in French le is used for masculine nouns (le jardin - the garden) and la for feminine nouns (la table - the table). The becomes les in front of plural nouns. The indefinite articles in French are un (masculine) and une (feminine). German, Dutch and Ancient Greek have masculine, feminine and neuter nouns, but in the case of Dutch the word for the is the same for masculine and feminine (de) so you do not need to know which it is. The Māori language uses one word for the for when the subject or object to which the the refers is one in number, and uses another word for the when the subject or object to which the the refers numbers more than one.
There are also many languages that do not have articles. Some examples include Russian, Hindi, Japanese, and Chinese. When speakers of these languages are learning English, it is often difficult to explain to them what an article is because English speakers use them naturally. Even though these languages do not have articles, they still have grammatical features that allow them to show definiteness.
In general: the in English is used for something you already know about or have already been talking about. The word a is used when introducing a new idea:
"The tired woman was looking for her cat. Suddenly she saw the cat up a tree". (We are already talking about the cat. The tree is a new idea).
"The tired woman was walking along when she suddenly saw a cat up a tree". (She had not been thinking about cats until then).
Sometimes we do not need an article, for example when talking about something in general:
"The dogs do not bite" (meaning: dogs that you are thinking about). "Barking dogs do not bite" (barking dogs in general).