There are seven days in a week. Each day is assigned to different Hindu deities. e.g. Monday for Shiva, Saturday for Hanuman, Thursday for Dattatreya, etc.
The day for Ganesha is Tuesday but fasting is not done for him on this day. The days to fast for Ganesha are Chaturthi days i.e. fourth day of each fortnight. The fourth day of the first fortnight in a month according to Hindu calendar is called as Vinayak Chaturthi. The fourth day of the second fortnight in a month according to Hindu calendar is called as Sankasht Chaturthi. If the Chaturthi comes on Tuesday, it is known as Angarak Chaturthi. The fast for Lord Ganesha is generally observed on Sankasht and Angarak Chaturthi. If you want to start fasting on Chaturthi, you should start it from any Angarak Chaturthi.
Besides those, you can observe fast for Ganesha on Ganesh Janma, Anant Chaturdashi, and during Ganesh festival, which is celebrated in the month of Bhadrapad. It is advised to visit Ganesha temple on Tuesdays and offer him a red flower if you are in some kind of problem.