Polysemy simply means many meanings, so one word has several definitions and grammatical functions. Homonymy is a related concept broken into two parts: homophones and homographs. Homophones are words that have the same pronunciation, whether or not they are spelled alike. Homographs are words that have the same spelling, but may or may not be pronounced alike. Homonyms are both homophones and homographs, i.e. words that are spelled and pronounced alike, but with different meanings. It is important to learn the different spellings, pronunciations and meanings of words in the beginning stage of language learning or you could say something you don’t mean or understand something to be what it is not.
Polysemy of common verbs is probably something that should be learned early on. Verbs such as faire, mettre, and passer have numerous translations in English depending on how they are used and with which expressions. Faire doesn’t always mean simply to do or to make, as it is also translated as to be (especially when talking about the weather) and other various expressions such as to get used to (se faire à) or to worry (s’en faire). The reflexive pronoun se and the prepositions that follow verbs can also completely change the meaning. Plaindre is to pity, but se plaindre is to complain. S’ennuyer is to get bored, but s’ennuyer de is to miss.
Homonyms are words like mer, mère, and maire. All of these nouns are pronounced the same so you really need to understand the context of the sentence or you won’t know whether someone is talking about the sea, a mother or a mayor. The word gare in French is most often first learned as train station. It is very basic vocabulary that all beginners know. But gare also means something else. When it is an exclamation instead of a noun, it means watch out or be careful. Another homonym is bois. As a noun it means wood, but as a verb, it is the singular conjugation for the verb drink. Cours is a lesson, class or course (among others), but the verb form cours means run, race or compete.
Examples like these are easier to spot in writing, since nouns generally need an article in front of them, but in everyday speech if you cannot understand every word and only catch certain basic vocabulary, you could completely misunderstand the message. This is also why learning the pronunciation of conjugations of verbs is important at the beginning stage. French has a large number of homonyms between nouns and verbs because of the numerous conjugations so it is not enough to just focus on the nouns or adjectives that sound alike.
Obviously for words that are spelled the same, it easy to look up their definition(s) and pronunciation(s) in the dictionary. But for words that are not spelled the same, yet are pronounced the same, it can be a bit trickier. Luckily the Dictionnaire du Francais that I posted about a few days ago does include notes about words that sound alike, and good vocabulary books include sections on homonyms, such as Vocabulaire expliqué du français (unfortunately, it is not available through Amazon.com)