The system on the right of the opening page can be used to choose the correct article in the most common situations. Please note, however, that the use of articles (or no article) is a very complex aspect of English grammar. In fact, whole books have been written on the topic (for example, the Collins Cobuild English Guides, Volume 3). For this reason, you should understand that the suggested article is most probably the right article for the chosen noun in your context (sentence). There is no computerized system, however, that will always choose the correct article for you. (If you think the answer suggested is wrong, send me an and I will advise you on your specific question.)
The most important thing to know in order to choose the correct article for the noun you want to use is to know whether the noun is a countable or uncountable noun. A countable noun is a noun that you can put a number in front of: 1 book, 2 books, 20 books, etc. An uncountable noun is a noun that you cannot put a number in front of. You cannot say 1 water, 2 waters, 20 waters, etc.
There are some nouns which can be either countable or uncountable, depending on their meaning in the context (sentence) in which they are being used. The noun hair is a good example of a noun that is both count and uncount. When it means all the hair on the head, hair is an uncount noun. This means that we cannot put a number in front of it or make it plural. So She has nice hairs is wrong. It has to be She has nice hair or, another example, Some cats don't have hair.
On the other hand, when hair is referring to the individual strands of hair, it is a count noun. It can therefore have a number in front of it. For example, There is one hair growing from the birthmark on his shoulder or Waiter, there are 3 hairs in my soup.
Another example is the noun paper. It is an uncount noun when it means the material that many things are made of: The plates in the cafeteria are made of paper. But when the word paper means a piece of academic writing or is short for newspaper, then it is count. I always buy a paper on the way home or She has written many papers on the topic of the banking system.
Here is a good, more detailed explanation of the article system.
Take a look around you. Everything you can see and name is a noun: computer, door, light, chair, etc. Nouns are also the names of feelings and concepts: sadness, health, democracy, etc.. And nouns are the names of people and places: Lincoln, Moscow, Everest, etc.
There are different ways to categorize nouns. One way is to divide them into common nouns such as water, computer, happiness) and proper nouns such as London, John or Christmas - note that all proper nouns start with a capital letter. Another way is to divide them into concrete nouns such as dog, river, cloud, and abstract nouns such as idea, problem and memory. A third way is according to whether a noun is count (it can be counted) or uncount (it cannot be counted). It is the count/uncount categorization that is the basis for deciding which, if any, article is needed.