Do the Christians or Jews fast and how?

Jews certainly do fast. There are several fast days throughout the year. Jews fast by not eating. (That's what "fast" means.)

On the major fast days, Jews also don't wash, wear makeup, have sex, or wear leather shoes. The minor fasts don't have these additional prohibitions.

I have a medical condition which dictates that I cannot do a complete fast. So I do a "Daniel" fast.

The concept of a Daniel fast comes from Daniel 1:8-14, “But Daniel resolved not to defile himself with the royal food and wine, and he asked the chief official for permission not to defile himself this way. Now God had caused the official to show favor and sympathy to Daniel, but the official told Daniel, ‘I am afraid of my lord the king, who has assigned your food and drink. Why should he see you looking worse than the other young men your age? The king would then have my head because of you.’

Daniel then said to the guard whom the chief official had appointed over Daniel, Hananiah, Mishael and Azariah, ‘Please test your servants for ten days: Give us nothing but vegetables to eat and water to drink. Then compare our appearance with that of the young men who eat the royal food, and treat your servants in accordance with what you see.’ So he agreed to this and tested them for ten days.”

The background of the Daniel fast is that Daniel and his three friends had been deported to Babylon when Nebuchadnezzar and the Babylonians conquered Judah (2 Kings 24:13-14). Daniel and his three friends were put into the Babylonian court servant “training program.” Part of the program was learning Babylonian customs, beliefs, laws, and practices. The eating habits of the Babylonians were not in complete agreement with the Mosaic Law. As a result, Daniel asked if he and his three friends could be excused from eating the meat (which was likely sacrificed to Babylonian false gods and idols).

So, a Daniel fast is eating only fruits and vegetables for a certain amount of time and abstaining from meat products.. It is a matter of Christian freedom whether to observe a Daniel fast.

It Depends If Like God Tells You To Or If You Just Want To Do It For A Spiritual Reason.