Cookies can be divided into two types. First-party cookies are cookies set by the website in the URL bar at the top of the page.
From the early days of the internet, it became obvious that it would be advantageous for websites to be able to store information on a computer for future access. For example, a website that displays weather information could ask the user for a zip code, and then store it in a cookie. The next time the user visited the website, weather information would automatically load for that zip code, without the user having to enter the zip code, and without the need for the user to create an account on the website (which would be overkill for such a simple task).
Some websites with logins require first-party cookies to be enabled for a user to stay logged in. Cookies aren't the only only way a website can maintain a user logged in as they move from page to page on the site, but if a particular website has chosen to implement logins in that way, enabling first-party cookies on that site will be the only way to use the functionality.
Third-party cookies are set by portions of a website that are loaded from servers different from the URL at the top of the page. For example, most website that have advertisements load them from a third-party ad broker, like Google's Ad Sense. Every time the website loads, it requests the ad broker to display some ads. The ad broker analyzes any information they may have about the user, looks at the current rate advertisers are willing to pay for their ads, and selects those to display. The section of the website that displays the ads is loaded from the third-party broker's server instead of the main server.
Because most of the advertisements on the internet are displayed from only a few brokers, it didn't take long for them to realize that they could set a tracking cookie on the user's device and know every place that user goes. Every time an ad loads from a broker, the first thing it does it check to see if if the device already has a unique serial number in a tracking cookie. If it does, it looks up the profile for that serial number and makes a note of the new site. This is why a user can do a search on one website for a product that they typically don't look for, like walnuts, and then suddenly start seeing advertisements for walnuts on every website they visit.
In addition to ad brokers, social media sites discovered they could get in on the action. A few years ago, the major social media sites like Facebook and Twitter convinced a large number of websites that it would be in there best interest to place little social media icons on their pages. These are not just images. They contain imbedded code that links back to the social media site, and, among other things, loads a third-party cookie on the device. These cookies are placed even if the user does not have an account with the social media platform. Over time, companies like Facebook (which also run an ad network) have built up quite a large number of detailed profiles about people who have never even created an account on their site.
There is almost no good reason to ever enable third-party cookies. On devices with Android KitKat or older (version <= 4.4.4 or API <= 20), WebView does not differentiate between first-party and third-party cookies. Thus, enabling first-party cookies will also enable third-party cookies.
Form data contains information typed into web forms, like user names, addresses, phone numbers, etc., and lists them in a drop-down box on future visits. Unlike the other forms of local storage, form data is not sent to the web server without specific user interaction.