As a professional editor, many if not most of my changes involve removing or adding commas. Since punctuation use is different for each language, I’ll try to provide a rule of thumb for Engish.
Without getting too technical, I can give you two approaches to sensing whether a comma isnecessary or not, one for the intuitive English learners and one for those who like short, simple rules.
A comma is the written equivalent of a pause in speech. We pause for good reasons, not randomly. We might want to add interesting, but not necessary information, as in “Sally’s cousin … the one that just took the bar exam … is coming to visit.” We stop to distinguish an incomplete or linked idea from the main idea: I’ll let you get on the bus … even if you don’t have the fare. A pause is used to communicate the direct connection between a name and title: Mr. Jones … the current CEO … has just received a bonus. By contrast, we don’t pause between the important parts of the sentence, unless we are trying to build suspense. My … new… job … pays… 10,000 dollars … a week sounds like someone has a speech impediment or has run up 12 flights of stairs. We also don’t pause if the description is required to understand the sentence. My neighbor with the AK17 in the living room is angry at me tells me that the rest of the neighborhood is not so bad. I do have other neighbors, still. What does this mean for comma use? If you would naturally stop to breathe, place a comma. If not, avoid a comma.
For those who don’t trust their ears and like hard and fast rules. Don’t place a comma between the subject and verb and the verb and objects. If the information is subordinate or extra, insert a comma.
By the way, American and some British usage places a comma before the and in a list, i.e. cats, dogs, and parrots. However, it is not a mistake to omit it.