"There is a bowl on the table."
I am wondering why 碗 doesn't require a measure word here. Is it because 碗 itself is used as a measure word at times?
It's because 碗 here isn't being "counted". It's just a noun. Chinese does not require articles as English does. In English you can't say "there is bowl," you have to say "there is a bowl". In Chinese, it's OK to say the equivalent of "there is bowl". You can use "一个碗“...if you want to specify that there is only 1 bowl, not 2 bowls, in which case you are "counting" it and so you need a measure word （个）. 碗 itself is a measure word when you are counting "bowls of" something. For example: "三碗米饭" is "three bowls of rice" whereas "三个碗" is "three bowls".