Millésime is a French word that translates reasonably well into English as 'vintage'. It is used in reference to wines, champagnes, and etc. English people think it's way cooler to say 'millésime' than 'vintage'.
To make things a bit more confusing, our word vintage would translate into French as vendange. As you may have guessed, they both come from the same root, as does the Spanish vendima and the Itallian vendemmia. English is the only one of these languages that considers 'vintage' to be the same thing as 'year'. In French, millésime refers to the vintage year, and vendange to the specific harvest. This is important because in some cases some grapes are left on the vine and picked late in the year (vendage tardive, 'late harvest'), giving two very different wines from the same year. Saying that a wine is a millésime may imply that it is a mixture of harvests produced that year.
Pronounced (roughly) 'mee-lay-zeem'*. Millésime is the noun (ex. Voici tous les millésimes. - "here are the vintages"); the adjective is millésimé, pronounced 'mee-leh-zeem-ay'. (ex. Vins non millésimé - "wines without year", i.e., a mix of wines from different years). Plural is millésimes.
* Disclaimer: my French is awful. Do not rely on my pronunciation if you want to sound sophisticated.