"Becha" is a Yiddish term for the special cup that Jews use for making Kiddush. The "ch" is pronounced as in the Scottish Loch.
Briefly, the Kiddush prayer is the Bracha over wine (or other drink) and associated Brachot for the day, that is recited at the evening and lunch meals on Shabbat and Yom Tov. Although wine is normally used, grape juice can be used instead, and at the lunch Kiddush (which is less important than the evening one), any drink can be used except water.
Most commonly, a Becha is made of silver and often decorated with engravings. However, there are also beautiful ones made of frosted glass, decorated china or anything else. All that said, there's nothing to stop you making the blessings over wine in a normal glass (or even a plastic disposable cup). But in keeping with the holiness of the Kiddush and "beautifying the Mitzvah", it is appropriate to use something more fancy.
Children may have their own small ones, which relatives often give as presents to a child on his or her birth, and may be engraved with their name in Hebrew or English, and the date of birth. Often a boy on his Bar Mitzvah will get a proper "large" Becha as a present, as it is then he is obliged to hear or recite Kiddush himself. And Synagogues sometimes give a couple a Becha on their marriage.