I'm working on a similar problem for use in machine learning, so let me share what I've been able to do on the topic.
1) If you know before you start parsing the row - whether it's hard-coded into your program or whether you've got some header in your file that gives you this information (highly recommended) - how many attributes per row there will be, you can reasonably split it by comma, for example the first attribute will be RowString.substring(0, RowString.indexOf(',')), the second attribute will be the substring from the first comma to the next comma (writing a function to find the nth instance of a comma, or simply chopping off bits of the string as you go through it, should be fairly trivial), and the last attribute will be RowString.substring(RowString.lastIndexOf(','), RowString.length()). The String class's methods are your friends here.
2) If you are having trouble distinguishing between commas which are meant to separate values, and commas which are part of a string-formatted attribute, then (if the file is small enough to reformat by hand) do what Java does - represent characters with special meaning that are inside of strings with '\,' rather than just ','. That way you can search for the index of ',' and not '\,' so that you will have some way of distinguishing your characters.
3) As an alternative to 2), CSVs (in my opinion) aren't great for strings, which often include commas. There is no real common format to CSVs, so why not make them colon-separated-values, or dash-separated-values, or even triple-ampersand-separated-values? The point of separating values with commas is to make it easy to tell them apart, and if commas don't do the job there's no reason to keep them. Again, this applies only if your file is small enough to edit by hand.
4) Looking at your file for more than just the format, it becomes apparent that you can't do it by hand. Additionally, it would appear that some strings are surrounded by triple double quotes ("""string""") and some are surrounded by single double quotes ("string"). If I had to guess, I would say that anything included in a quotes is a single attribute - there are, for example, no pairs of quotes that start in one attribute and end in another. So I would say that you could:
Make a class with a method to break a string into each comma-separated fields.
Write that method such that it ignores commas preceded by an odd number of double quotes (this way, if the quote-pair hasn't been closed, it knows that it's inside a string and that the comma is not a value separator). This strategy, however, fails if the creator of your file did something like enclose some strings in double double quotes (""string""), so you may need a more comprehensive approach.