The applications start usually in January and end in June-July. The more specific schedule depends on the institute you are applying for. Most applications take place on Parcoursup, whereas some take place directly to the institute in question. It is strongly recommended to find out about the application procedure in place well in advance. 

By way of example, here is the Parcoursup schedule for 2020:

Parcoursup

• January 22nd: the platform is opened online 
• April 2nd: the end of the first phase
• May 19th: the institutes' decisions are communicated
• July 17th: the last chance to accept an offered place of study